#groups-rings-fields | Mathematics | Page 423

oblique river Jan 23, 2019, 1:23 AM

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maybe I dont understand the question, sorry

daring wolf Jan 23, 2019, 1:24 AM

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how many generators does the field have

oblique river Jan 23, 2019, 1:24 AM

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oh, yeah that's the question I was answering

daring wolf Jan 23, 2019, 1:24 AM

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oh no no it's most likely that i didn't explain well enough

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or i just don't get it

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oh hmmm

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why does the rest of the 121-11 generate all of the F_121

oblique river Jan 23, 2019, 1:25 AM

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because they have to generate something

daring wolf Jan 23, 2019, 1:25 AM

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and how does the quadratic tie into this 030

oblique river Jan 23, 2019, 1:25 AM

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the quadratic doesn't really matter at all

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tbh

daring wolf Jan 23, 2019, 1:25 AM

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hmmmm

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so it'd have 121-11 generators o3o?

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I don't quite understand why though

oblique river Jan 23, 2019, 1:26 AM

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it doesn't make sense to talk about "generators of a quadratic"

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a generator of a field is an element x such that linear combinations of powers of x give you everything

daring wolf Jan 23, 2019, 1:26 AM

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yup i get that part

oblique river Jan 23, 2019, 1:26 AM

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so in this case, let's call t a root of that polynomial

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then you can write F_121 as the set of all a + bt where a and b are in F11

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but you don't need to do that for this problem

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it's true that in fact there is a unique field of order 121, and there are several ways to write it. Maybe the reason they're including the polynomial is so that they can "pin down" a "specific" copy of F_121 since they haven't proven that there is a unique field of order 121 that they can just call F_121

daring wolf Jan 23, 2019, 1:28 AM

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well there is more to the question too i guess

oblique river Jan 23, 2019, 1:28 AM

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I mean, it's possible to write a proof for this problem that uses the polynomial in an explicit way. But it's just not necessary to do so.

daring wolf Jan 23, 2019, 1:28 AM

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Find a field element of order 4 and 5

oblique river Jan 23, 2019, 1:28 AM

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oh. yeah then you do want to use that polynomial

daring wolf Jan 23, 2019, 1:28 AM

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after that first question

oblique river Jan 23, 2019, 1:28 AM

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for that part

daring wolf Jan 23, 2019, 1:29 AM

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ok wait so how did you get 121-11 generators

oblique river Jan 23, 2019, 1:29 AM

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so

daring wolf Jan 23, 2019, 1:29 AM

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why is it not 121

oblique river Jan 23, 2019, 1:29 AM

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pick 0

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does 0 generate F_121?

daring wolf Jan 23, 2019, 1:29 AM

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granted 0 is not a generator

oblique river Jan 23, 2019, 1:29 AM

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no, because all the powers of 0 are just 0

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so you're already down to 120

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now pick an element of F_11

daring wolf Jan 23, 2019, 1:29 AM

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120 more to go o3o

oblique river Jan 23, 2019, 1:29 AM

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all of its powers are in F_11

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and so all of the sums of those powers are in F_11

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so it generates F_11

daring wolf Jan 23, 2019, 1:30 AM

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but shouldn't it generate F_121 too?

oblique river Jan 23, 2019, 1:30 AM

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how would it?

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the element t = 0 + 1*t is in F_121 (same t as before)

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how are you gonna get that t if you start with the element 2?

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all you're allowed to do is take powers and multiply by things in F_11

daring wolf Jan 23, 2019, 1:31 AM

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wait

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elements are the coefficients?

oblique river Jan 23, 2019, 1:31 AM

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okay

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If L and K are fields

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and L contains K

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then a generator for L over K

daring wolf Jan 23, 2019, 1:31 AM

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sorry i think i am missing some knowledge about Fields

oblique river Jan 23, 2019, 1:31 AM

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is some element x in L

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such that every element of L

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can be obtained using addition, multiplication, and multiplying by elements of K

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and that x

daring wolf Jan 23, 2019, 1:32 AM

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what does "a generator for L over K" mean

oblique river Jan 23, 2019, 1:32 AM

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I just told you :)

daring wolf Jan 23, 2019, 1:32 AM

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sorry >< I am really lacking volcabularies

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oh okokhmmm

oblique river Jan 23, 2019, 1:33 AM

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yeah it's gonna be hard to do this problem about generators of a field extension if you aren't familiar with what a generator of a field extension is lol

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here is another way to think about it

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an element x of L is a generator for L over K if

daring wolf Jan 23, 2019, 1:33 AM

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cuz yea i understand what a generator in a cyclic group is XD

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but not so much here

oblique river Jan 23, 2019, 1:33 AM

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L is the smallest field containing both x and K

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it's the same idea

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a generator for a group is "an element such that you get everything in the group by repeatedly doing group operations"

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so a generator for a field is "an element such that you get everything in the field by repeatedly doing field operations"

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but if you talk about a generator for L over K

daring wolf Jan 23, 2019, 1:34 AM

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yea so in this case + or x

oblique river Jan 23, 2019, 1:34 AM

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you get to include things in K in that as well

daring wolf Jan 23, 2019, 1:35 AM

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so the generator in L generates everything in K too?

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is that what that means?

oblique river Jan 23, 2019, 1:35 AM

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no

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you get the stuff in K for free

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the question is

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"can you get all of L just starting from K and x?"

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if yes, then x is a generator for L over K

daring wolf Jan 23, 2019, 1:35 AM

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OH

oblique river Jan 23, 2019, 1:35 AM

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if not, then x is not a generator for L over K

daring wolf Jan 23, 2019, 1:36 AM

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hmmmm

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I think i got it for a second and lost it again...

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so x is the generator for both K and L

oblique river Jan 23, 2019, 1:36 AM

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no

daring wolf Jan 23, 2019, 1:36 AM

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Sorry ><

oblique river Jan 23, 2019, 1:36 AM

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"L over K" is a single unit

daring wolf Jan 23, 2019, 1:37 AM

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wait

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what

oblique river Jan 23, 2019, 1:37 AM

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x is a generator for (L over K) if you can get all of L, starting from just x and K

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x is probably not going to be in K

daring wolf Jan 23, 2019, 1:37 AM

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L/K?

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is that the notation?

oblique river Jan 23, 2019, 1:37 AM

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yes

daring wolf Jan 23, 2019, 1:37 AM

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oh...

oblique river Jan 23, 2019, 1:37 AM

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I didn't wanna write that cuz I didn't want to confuse you because that looks like a quotient

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but it's not a quotient

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it's the notation for "L is a field extension of K"

daring wolf Jan 23, 2019, 1:38 AM

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so like in my question it'd be F_121/Z_11[x]?

oblique river Jan 23, 2019, 1:38 AM

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no

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Z_11[x] is not a field

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and also it's massive, it's not contained in F_121

daring wolf Jan 23, 2019, 1:38 AM

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wait K is also a field?

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L is a field

oblique river Jan 23, 2019, 1:38 AM

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I mean a field extension means that both are fields lol

daring wolf Jan 23, 2019, 1:38 AM

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ah icic

oblique river Jan 23, 2019, 1:38 AM

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L = F_121 and K = F_11

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(by F_11 I mean the field of 11 elements, which you called Z_11, I just prefer to call it F_11 to emphasize that it is a field)

daring wolf Jan 23, 2019, 1:39 AM

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ah icic

oblique river Jan 23, 2019, 1:39 AM

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so let's go back to you rproblem

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you want to find all the generators of F_121 over F_11

daring wolf Jan 23, 2019, 1:40 AM

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yea

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or well the total number

oblique river Jan 23, 2019, 1:40 AM

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so pick an element of F_121

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let's say that element happened to be in F_11

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then you're never going to escape F_11

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if all you're allowed to do is "field things + F_11"

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because F_11 is already closed under field things!

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It's like asking "what happens if you start with the integers Z and add in the number 7?"

daring wolf Jan 23, 2019, 1:41 AM

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oh it's still gonna be in Z

oblique river Jan 23, 2019, 1:41 AM

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you just get Z back because you didn't even add anything

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exactly

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same thing here

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if you start iwht F_11 and throw in something that was already in F_11

daring wolf Jan 23, 2019, 1:41 AM

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so if i add things to this F_11 element

oblique river Jan 23, 2019, 1:41 AM

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you're still F_11

daring wolf Jan 23, 2019, 1:41 AM

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i'd still get things in F_11

oblique river Jan 23, 2019, 1:41 AM

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if you add things in F_11 to it, then yes

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and remember, that's what's important here

daring wolf Jan 23, 2019, 1:41 AM

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yup got that part XD

oblique river Jan 23, 2019, 1:41 AM

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ok great

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so now we're down to 110 possibilities

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we ruled out all 11 elements of F_11

daring wolf Jan 23, 2019, 1:42 AM

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ok so this x in both F_121 and F_11 is a generator for F_121

oblique river Jan 23, 2019, 1:42 AM

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no

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if x is in F_11

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then it is not a generator of F_121

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F_11 is a subset of F_121

daring wolf Jan 23, 2019, 1:42 AM

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since it goes back to F_11

oblique river Jan 23, 2019, 1:42 AM

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yes, because what you're doing is

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"start iwth F_11 and toss in that x"

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if x was already in F_11, you just did nothing

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and you still have F_11

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but now... what if that x was not in F_11?

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(of course it's still in F_121, because that's the scope of this problem)

daring wolf Jan 23, 2019, 1:43 AM

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then it can or cannot be a generator?

oblique river Jan 23, 2019, 1:43 AM

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yes

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I claim that in fact it must be a generator of F_121

daring wolf Jan 23, 2019, 1:43 AM

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we would have 121-11 possible generators

oblique river Jan 23, 2019, 1:44 AM

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so that in fact the answer is 121-11

daring wolf Jan 23, 2019, 1:44 AM

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why must it be though?

oblique river Jan 23, 2019, 1:44 AM

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take that x

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it generates something

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like, if you start with F_11 and throw in x

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you get something

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but that something must be strictly bigger than F_11

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because you just added in x, which wasn't in F_11

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but, I claim that there is nothing in between F_11 and F_121!

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so as soon as you're bigger than F_11, you must be all of F_121

daring wolf Jan 23, 2019, 1:45 AM

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wait F_11 is a subset of F_121 but an element in F_11 is not in F_121?

oblique river Jan 23, 2019, 1:45 AM

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no no no

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our element x is not in F_11

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F_11 has 11 elements

daring wolf Jan 23, 2019, 1:45 AM

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oh

oblique river Jan 23, 2019, 1:45 AM

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we already dealt with those

daring wolf Jan 23, 2019, 1:46 AM

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a different x

oblique river Jan 23, 2019, 1:46 AM

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remember? that was the 11 in 121-11

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our x is a random element of F_121

daring wolf Jan 23, 2019, 1:46 AM

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yea okok XD i was confused haha my bad

oblique river Jan 23, 2019, 1:46 AM

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either that element x is in F_11

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or it is not in F_11

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we already dealt with the "in F_11" case

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so now we've moved on to the "not in F_11" case

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if you want to prove that there's nothing in between F_11 and F_121

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well, there are several ways

daring wolf Jan 23, 2019, 1:47 AM

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but the non F_11 elements do they all generate all of the F_121 ?

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like i don't understand why would they generate all of F_121 elements

oblique river Jan 23, 2019, 1:47 AM

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that's what we're proving...

daring wolf Jan 23, 2019, 1:48 AM

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oh okok

oblique river Jan 23, 2019, 1:48 AM

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sorry, I think we've lost each other

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what do you think I'm trying to do now?

daring wolf Jan 23, 2019, 1:49 AM

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prove that there is no other fields between F_11 and 121?

oblique river Jan 23, 2019, 1:49 AM

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yeah but why was I trying to do that

daring wolf Jan 23, 2019, 1:49 AM

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that I have no idea ><

oblique river Jan 23, 2019, 1:49 AM

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because

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our element x

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generates some field between F_11 and F_121

daring wolf Jan 23, 2019, 1:50 AM

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wait what

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how does it do that

oblique river Jan 23, 2019, 1:50 AM

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what do you mean?

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start with x and F_11

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do all the field things you can to them

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you're gonna get a field

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that's by definition what "field generated by" is

daring wolf Jan 23, 2019, 1:50 AM

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ok

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so let's say a is an element of F_11

oblique river Jan 23, 2019, 1:51 AM

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ok

daring wolf Jan 23, 2019, 1:51 AM

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so we are making x*a and x+a

oblique river Jan 23, 2019, 1:51 AM

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and powers of x

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and sums and products and inverses

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of all those things

daring wolf Jan 23, 2019, 1:51 AM

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and that gives us a new field

oblique river Jan 23, 2019, 1:51 AM

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yes

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that field certainly contains F_11, since I let myself have that for free

daring wolf Jan 23, 2019, 1:52 AM

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wait is F_121 containing a bunch of other fields?

oblique river Jan 23, 2019, 1:52 AM

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and it certainly is contained in F_121 since both x and F_11 are contained in F_121, so doing field things certainly can't leave F_121

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I mean

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no

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the only subfield of F_121 is F_11

daring wolf Jan 23, 2019, 1:52 AM

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how is F_11 in F_121 then

oblique river Jan 23, 2019, 1:53 AM

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what?

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F_121 was given to you as "the field you get from F_11 and the root ofsome quadratic polynomial"

daring wolf Jan 23, 2019, 1:53 AM

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i think i am starting to understand certain parts but I am getting confused about F_11 is contained in F_121

oblique river Jan 23, 2019, 1:53 AM

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you don't just lose F_11 when yo do that

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okay

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you know how the field R is contained in the field C?

daring wolf Jan 23, 2019, 1:54 AM

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yes

oblique river Jan 23, 2019, 1:54 AM

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or Q is contained in Q(i) = {a + bi | a, b in Q}

daring wolf Jan 23, 2019, 1:54 AM

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wait

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Reals are contained in Complex?

oblique river Jan 23, 2019, 1:54 AM

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wait

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is that a joke?

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complex numbers are things of the form a + bi

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where a and b are real

daring wolf Jan 23, 2019, 1:54 AM

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ahh i am sorry ><

oblique river Jan 23, 2019, 1:54 AM

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any real number is of the form a + 0i

daring wolf Jan 23, 2019, 1:55 AM

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... wow

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i suck...

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ok got that ...

oblique river Jan 23, 2019, 1:55 AM

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well

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what I was gonna say

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was that just how R is contained in C

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and Q is contained in Q(i)

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F_11 is contained in F_121

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like Q(i) is "start with Q, throw in i, and do all the field things you want"

daring wolf Jan 23, 2019, 1:55 AM

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k i think i got that part now

oblique river Jan 23, 2019, 1:56 AM

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F_121 is "start with F_11, add in a root of that quadratic polynomial, adn do all the field things you want"

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clearly F_11 is in there

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because that's what you started with

daring wolf Jan 23, 2019, 1:57 AM

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ok so we do field things to F_11

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through x that belongs to F_121

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then we start generating things in F_121

oblique river Jan 23, 2019, 1:58 AM

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yes

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so the question is

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if we start with some random x

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do we get all of F_121?

daring wolf Jan 23, 2019, 1:58 AM

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yea that XD

oblique river Jan 23, 2019, 1:58 AM

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or just some proper subfield of F_121?

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so like let's go back to the R example

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R inside C

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i is a generator for C over R

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because if you start with R, and add in i

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you get all of C

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but sqrt(2) is not a generator for C over R

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because if you add in sqrt(2) to R

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you just get R back

daring wolf Jan 23, 2019, 1:59 AM

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yup

oblique river Jan 23, 2019, 1:59 AM

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because sqrt(2) was in R to begin with

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let's keep thinking about this example

daring wolf Jan 23, 2019, 1:59 AM

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a+bi

oblique river Jan 23, 2019, 1:59 AM

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let's say I give you some complex number a+bi

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when does that generate C over R?

daring wolf Jan 23, 2019, 2:00 AM

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when a and b belongs to R?

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0.0

oblique river Jan 23, 2019, 2:01 AM

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no

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we just said that sqrt(2) = sqrt(2) + 0i does not generate C over R

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and more generally

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we said that any real number obviously can't generate C over R

daring wolf Jan 23, 2019, 2:02 AM

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b>0

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or well b=/=0

oblique river Jan 23, 2019, 2:02 AM

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because if you "start with R and add in a real number" then you just get R back

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yes!

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if you take any element in C

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that's not in R

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then it generates C over R

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here's another question: Does i generate C over Q?

daring wolf Jan 23, 2019, 2:04 AM

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no?

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since it cannot cover all of C?

oblique river Jan 23, 2019, 2:04 AM

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can you give me an explicit element of C

daring wolf Jan 23, 2019, 2:04 AM

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0.5+0.5i

oblique river Jan 23, 2019, 2:04 AM

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that you can't get by just "starting with Q and tossing in i"?

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that's a bad example cuz you can ger that one lol

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that's (1 + i)/2

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haha you gotta wait until I finish asking the question before you try to answer :P

daring wolf Jan 23, 2019, 2:05 AM

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pi+pi i?

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yea my bad

oblique river Jan 23, 2019, 2:05 AM

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yes, but an even "easier" answer would just be pi

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you can't get pi just starting from Q and i!

daring wolf Jan 23, 2019, 2:05 AM

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oh?

oblique river Jan 23, 2019, 2:05 AM

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I mean, if you could get pi

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then you could clearly get pi + pi i

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because pi + pi i = pi(1 + i)

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and if you can get pi, and if you can get 1 + i, then you can get pi(1+i)

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because you multiply them

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how would you propose getting pi from Q and i?

daring wolf Jan 23, 2019, 2:06 AM

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maybe i am not understanding the concept of "over"

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I thought in some context it's kinda like modulo

oblique river Jan 23, 2019, 2:06 AM

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no no no no

daring wolf Jan 23, 2019, 2:06 AM

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and in some context it is extension

oblique river Jan 23, 2019, 2:06 AM

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that is why the notation L/K is misleading

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there are no quotients

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no modulo

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forget everything you know about the notation

daring wolf Jan 23, 2019, 2:07 AM

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oh ok

oblique river Jan 23, 2019, 2:07 AM

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the only definition you can work with right now is

daring wolf Jan 23, 2019, 2:07 AM

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what is over exactly then XD

oblique river Jan 23, 2019, 2:07 AM

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writing L/K just means that L is a field extension of K

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that's it

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so I woudl write "i is a generator of C/R"

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because C is a field extension of R

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and you can get all of C

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by starting iwth R and throwing in i

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similarly, i is a generator for Q(i)/Q

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but i is not a generator for C/Q

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because if you start with Q

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and throw in i

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you are never going to get pi

daring wolf Jan 23, 2019, 2:09 AM

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field extension means R is a subfield of C right?

oblique river Jan 23, 2019, 2:09 AM

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yes

daring wolf Jan 23, 2019, 2:09 AM

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just in case i get definitions wrong again

oblique river Jan 23, 2019, 2:09 AM

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that's it

daring wolf Jan 23, 2019, 2:10 AM

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ok so F_11 is a subfield of F_121

oblique river Jan 23, 2019, 2:10 AM

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yes

daring wolf Jan 23, 2019, 2:10 AM

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F_121 over F_11

oblique river Jan 23, 2019, 2:10 AM

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yes

daring wolf Jan 23, 2019, 2:10 AM

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and F_11 has 10 elements

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in mult

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and 11 in addition

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F_121 would have 121 elements

oblique river Jan 23, 2019, 2:11 AM

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yes

daring wolf Jan 23, 2019, 2:11 AM

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wait

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so we were talking about F_11 has 11 generators then?

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for F_11>

oblique river Jan 23, 2019, 2:11 AM

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no

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it has 10

daring wolf Jan 23, 2019, 2:11 AM

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ah yes

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sorry

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0

oblique river Jan 23, 2019, 2:11 AM

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well, I guess tha'ts kinda a technicality

daring wolf Jan 23, 2019, 2:12 AM

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in mult it doesn't exist

oblique river Jan 23, 2019, 2:12 AM

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like I guess you could say that 0 generates F_11 over F_11

daring wolf Jan 23, 2019, 2:12 AM

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nor in add

oblique river Jan 23, 2019, 2:12 AM

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because if you "start with F_11 and add in 0" you still get F_11

daring wolf Jan 23, 2019, 2:12 AM

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true..

oblique river Jan 23, 2019, 2:12 AM

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but in any case it doesn't matter

daring wolf Jan 23, 2019, 2:12 AM

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hmmmm

oblique river Jan 23, 2019, 2:12 AM

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if you start with any element of F_11

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you're never going to escape F_11

daring wolf Jan 23, 2019, 2:12 AM

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ok so now we know that the elements in F_11 is not a generator in F_121

oblique river Jan 23, 2019, 2:12 AM

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just like with the sqrt(2) in C/R. starting with R and tossing in sqrt(2), yhou're never going to escape R

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correct

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so we're down to 121-11 possibilities

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I claim

daring wolf Jan 23, 2019, 2:13 AM

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AH icic

oblique river Jan 23, 2019, 2:13 AM

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that all of them generate F_121 over F_11

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in the same way that any non-real element of C generates C

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any non-F_11 element of F_121 will generate F_121

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to be clear we haven't proven that yet. I'm just telling you what we are going to prove.

daring wolf Jan 23, 2019, 2:14 AM

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wait doesn't an element have to be let's say a belongs to F_121, a^n, n belongs to Natural numbers, then a is a generator?

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and a^n generates all of F_121

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or do i have the definition of generator wrong

oblique river Jan 23, 2019, 2:14 AM

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wait

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is this problem electronic? can you screenshot it?

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it's super important if we're talking about "generator of F_121" or "generator of (F_121)*"

daring wolf Jan 23, 2019, 2:15 AM

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📎 unknown.png

oblique river Jan 23, 2019, 2:15 AM

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okay

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yeah we're good

daring wolf Jan 23, 2019, 2:15 AM

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oh the other one is finite field isn't it

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(F_121)*

oblique river Jan 23, 2019, 2:16 AM

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wait

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i'm just worried

daring wolf Jan 23, 2019, 2:16 AM

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i feel like i got the concept of generator wrong

oblique river Jan 23, 2019, 2:16 AM

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because asking for a generator of F_121*

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is a generator of the group of units

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which is different from a generator of the field

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is there a definition from earlier thatn you can screenshot?

daring wolf Jan 23, 2019, 2:16 AM

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i think i am wrong

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i am pretty sure this is talking about generator of the field

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and I think i have been viewing this wrong

oblique river Jan 23, 2019, 2:17 AM

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oh okay

daring wolf Jan 23, 2019, 2:17 AM

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wait

oblique river Jan 23, 2019, 2:17 AM

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you should double check because they are very differnet problems

daring wolf Jan 23, 2019, 2:17 AM

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so what is the generator of a field

oblique river Jan 23, 2019, 2:17 AM

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that's what we've been talking about this whole tiem

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the other one is a generator of the group of units

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let's just pretend we're workign with what we have been this whole time

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cuz that's what it looks like and also I gotta go soon lol

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actually I have to go like very soon

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i'll just try to go through everything one more time very quickly

daring wolf Jan 23, 2019, 2:19 AM

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oh sorry ><

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thanks anyways

oblique river Jan 23, 2019, 2:19 AM

#

a generator of F_121 over F_11 is an element x in F_121 such that you can get everything in F_121 by starting with F_11 and throwing in x and doing field things

#

take an x in F_121

#

if x is in F_11 then you can't ever escape F_11

#

so that won't work

#

now if x is not in F_11

#

it must generate some field

#

just start with F_11 and x and do all the field things

#

what you get is clearly bigger than F_11

#

cuz x wasn't in F_11 to begin with, so of course you're bigger than F_11

#

I claim that you get all of F_121.

#

like I said, you must generate some field

#

but there are no other fields between F_11 and F_121

daring wolf Jan 23, 2019, 2:20 AM

#

I think I get what you meant now lol

#

been looking at it in the units way

oblique river Jan 23, 2019, 2:20 AM

#

ohhhh

#

yeah haha that woudl explain the confusion

daring wolf Jan 23, 2019, 2:20 AM

#

by any chance you know where can i view this in the units way?

oblique river Jan 23, 2019, 2:21 AM

#

I mean it's just a completely different problem

#

but yes

#

there is a theorem (idk if you've proven it) that if F is a finite field

#

then the group of units is cyclic

#

so for F_121, the group of units is cycli

#

and of order 120 (cuz its everything but 0)

#

so the question is just

#

"how many generators does a cyclic group of order 120 have"

daring wolf Jan 23, 2019, 2:21 AM

#

wait phi(121-1)?

oblique river Jan 23, 2019, 2:21 AM

#

which is phi(120)

#

lol that version is much easier

daring wolf Jan 23, 2019, 2:22 AM

#

Euler's totient function

#

OMG

#

that

#

is what i actually am looking for....

#

XD

#

sorry for the confusion

oblique river Jan 23, 2019, 2:22 AM

#

but normally I think people would call that a "primitive element" of F_121

daring wolf Jan 23, 2019, 2:22 AM

#

yea

#

he also mentioned it once

oblique river Jan 23, 2019, 2:22 AM

#

whereas a "generator" of F_121 is what I was talking about this whole time

daring wolf Jan 23, 2019, 2:22 AM

#

that other people call it primitive element

oblique river Jan 23, 2019, 2:22 AM

#

rofl

#

but not really rofl

daring wolf Jan 23, 2019, 2:22 AM

#

FML

oblique river Jan 23, 2019, 2:22 AM

#

it's okay

daring wolf Jan 23, 2019, 2:22 AM

#

XD

oblique river Jan 23, 2019, 2:22 AM

#

i'm just going to tell myself

daring wolf Jan 23, 2019, 2:22 AM

#

wait

oblique river Jan 23, 2019, 2:23 AM

#

that you would ahve to learn this stuff eventually

daring wolf Jan 23, 2019, 2:23 AM

#

so do i just compute the Euler's totient function

#

for 121-1

oblique river Jan 23, 2019, 2:23 AM

#

so I didnt just waste both of our times

daring wolf Jan 23, 2019, 2:23 AM

#

but that's massive

oblique river Jan 23, 2019, 2:23 AM

#

avlkalkjalkda

daring wolf Jan 23, 2019, 2:23 AM

#

yea what you explained made massive sense to me

#

sounds more like a generator than what I have been learning as generator way more

#

and it clarified a lot of things for me

oblique river Jan 23, 2019, 2:23 AM

#

if you can factor 120 into primes then you can find its euler totient very ezpz

daring wolf Jan 23, 2019, 2:24 AM

#

i.e. R belongs to C

#

ah imma try that out now

#

just before you go

#

what does it mean by a field element of order 4 and 5

#

like what is it asking

#

just an element within F_121 with that quadratic equation that ^4 will result in 1?

oblique river Jan 23, 2019, 2:26 AM

#

yes

#

so like

#

something that "looks like i"

#

cuz i^4 = 1

#

you know everythign in F_121 is of the form a + bt where t is a root of that quadratic

#

so just play with numbers

#

until you get it to work

daring wolf Jan 23, 2019, 2:27 AM

#

yes sir 😄

daring wolf Jan 23, 2019, 3:34 AM

#

if anyone is still around

#

the multiplicative inverse of x in Z[x]/<3,x^3+2x+1>

#

does this multiplicative inverse have to be irreducible?

#

or it doesn't matter

#

o.o

daring wolf Jan 23, 2019, 4:20 AM

#

nvm figured it out

sonic current Jan 23, 2019, 2:51 PM

#

guys, I have a problem figuring out if a morphism is surjective

thorny slate Jan 23, 2019, 2:52 PM

#

ask

sonic current Jan 23, 2019, 2:52 PM

#

say we have a direct product

📎 Screenshot_2019-01-23_at_14.52.04.png

#

📎 Screenshot_2019-01-23_at_14.52.07.png

#

defined as such

#

I'm not sure whether all elements of (Z8,+)X(Z,+) must be represented in θ or just the ones in the form ([n]8,n)

#

do you know what I mean?

#

(5,3) is in (Z8,+)X(Z,+) but the morphism will never produce it

thorny slate Jan 23, 2019, 2:57 PM

#

yeah it's not surjective

#

to prove a morphism of finitely generated groups is surjective just check generators

#

(1,0) and (0,1) in this case

#

also don't write the direct product like that

#

that's the tensor product symbol

sonic current Jan 23, 2019, 2:58 PM

#

it's my teacher's notation

#

😦

#

so what could my argument be?

thorny slate Jan 23, 2019, 2:59 PM

#

note that the image is exactly (n mod 8, n)

#

so for example (0,1) isn't in the image

sonic current Jan 23, 2019, 2:59 PM

#

I instinctively thought about not every element of (Z8,+)X(Z,+) to be generated

#

ok yeah

#

my problem was if we would restrict surjectivity to the morphism span or something

#

but it makes sense! thanks

timber bay Jan 24, 2019, 12:15 AM

#

im not really sure how to start this. im kind of confused at what its asking

📎 unknown.png

#

i have Q[x]/(f(x)) where f(x) is given

#

I know for a, f(alpha^2) = 0, but im not sure how I should generate a polynomial out of this

#

im also given an alpha = x+(f(x)) in the first question

#

<@&286206848099549185>

thorny slate Jan 24, 2019, 12:42 AM

#

show the field maybe

timber bay Jan 24, 2019, 12:42 AM

#

ok

#

ive already finished this one

📎 unknown.png

thorny slate Jan 24, 2019, 12:47 AM

#

so you need the minimum polynomial of x^2

#

you said that f(x^2) = 0

#

if this is true, then since f is irreducible, it has to be the minimum polynomial

#

another way of computing minimum polynomials is by writing the elements as matrices

#

over the basis (1, x, x^2, x^3)

#

and finding the minimal polynomial of that matrix

#

using linear algebra

#

I would do that for the others

timber bay Jan 24, 2019, 12:52 AM

#

we haven't dealt with actual matrices in this class. I have taken some linear algebra before, but im gonna need to refresh my memory.

#

oh I meant that I need to find an f(x) so that f(alpha^2) = 0

#

the f(x) in the question doesnt satisfy this condition.

thorny slate Jan 24, 2019, 1:11 AM

#

oh

#

yeah it looks annoying if you haven't seen the linear algebra interpretation

#

are you supposed to just guess?

#

not sure

timber bay Jan 24, 2019, 2:11 AM

#

yeah me neither @thorny slate

north tide Jan 24, 2019, 6:03 AM

#

From minimal polynomial,\alpha(1+\alpha^2)=1+\alpha^2+\alpha^4, square both sides and we see \alpha^2 satisfies x(1+x)^2=(1+x+x^2)^2, which simplifies to 1+x+x^2+x^3+x^4=0. One can show that this is irreducible over rational polynomials because the factorization as in real polynomials contain irrational coefficients. @timber bay

sonic current Jan 24, 2019, 11:47 AM

#

hello everyone! quick question here. Could this be a field with the usual addition and multiplication in Z15?

📎 Screenshot_2019-01-24_at_11.46.33.png

#

no multiplicative identity

#

uhhh

#

nevermind

#

😓

bleak abyss Jan 24, 2019, 3:10 PM

#

Should be isomorphic to Z/3

thorny slate Jan 24, 2019, 3:50 PM

#

yeah

#

10 is 1

chilly ocean Jan 24, 2019, 5:29 PM

#

Can someone teach me higher mathematics?

lyric falcon Jan 24, 2019, 5:44 PM

#

What higher mathematics?

timber bay Jan 24, 2019, 8:13 PM

#

if you want to learn higher mathematics you must get higher then learn mathematics

worthy kindle Jan 24, 2019, 8:14 PM

#

school can teach you :p

solar wyvern Jan 24, 2019, 8:26 PM

#

@chilly ocean general questions like this go in the general channels #math-discussion #chill #discussion

timber bay Jan 24, 2019, 10:26 PM

#

i'm not exactly sure how to approach createing an element so that Q(x) is an n dimensional vector space

📎 unknown.png

#

not exactly sure what that would look like

bleak abyss Jan 24, 2019, 10:55 PM

#

Do you know the tower law?

#

If so, then you'll know that Q(sqrt(2), sqrt(3)) is 4-dimensional anyway

thorny slate Jan 24, 2019, 10:58 PM

#

it has to be an element that lets your recover both sqrt(2) and sqrt(3)

#

I'd try the "obvious" candidates and one of them just happens to work

#

can you do better than this @bleak abyss ?

#

or is that the idea

bleak abyss Jan 24, 2019, 10:59 PM

#

Pretty much

thorny slate Jan 24, 2019, 10:59 PM

#

Sad!

timber bay Jan 24, 2019, 11:24 PM

#

I do know the tower law. i'm not sure how this applies. @bleak abyss

thorny slate Jan 24, 2019, 11:24 PM

#

what's the tower law

timber bay Jan 24, 2019, 11:24 PM

#

i get that the basis of Q(sqrt(2), sqrt(3)) is 4 dimensional now, but not sure what sort of element alpha is

thorny slate Jan 24, 2019, 11:25 PM

#

you just need to find a degree 2 element

#

that should be simple, yeah?

bleak abyss Jan 24, 2019, 11:25 PM

#

You want an element which doesn't contain just the data of sqrt(2) or just of sqrt(3)

#

So mix them together somehow

#

What's the obvious candidate?

thorny slate Jan 24, 2019, 11:25 PM

#

we were discussing part (b) which is the hardest one

#

(a) and (c) are direct

timber bay Jan 24, 2019, 11:25 PM

#

maybe i know a different version? if a is an extension of b and b is an extension of c, and b has dimension x and c has dim y then a has dim xy

#

would a degree two element be like x^2 - 3

thorny slate Jan 24, 2019, 11:26 PM

#

that's sqrt(3) yeah

#

that's one

timber bay Jan 24, 2019, 11:26 PM

#

or x^2-2 or something

thorny slate Jan 24, 2019, 11:26 PM

#

and that's sqrt(2)

#

so any of those works

#

for (a)

timber bay Jan 24, 2019, 11:27 PM

#

why do they have a basis of size 2?

thorny slate Jan 24, 2019, 11:27 PM

#

oh you haven't worked through that?

#

let me show you

#

start with a field K

timber bay Jan 24, 2019, 11:28 PM

#

i might have

thorny slate Jan 24, 2019, 11:28 PM

#

as you know, we usually do a field extension by doing K[x]/p(x) where p is an irreducible polynomial in K[x]

timber bay Jan 24, 2019, 11:28 PM

#

i feel like my class is moving faster than id like tbh

thorny slate Jan 24, 2019, 11:28 PM

#

if the degree of p is d,then this is going to be an extension of degree d

timber bay Jan 24, 2019, 11:28 PM

#

ohhhhh

#

yeah i got it now

thorny slate Jan 24, 2019, 11:28 PM

#

and it's going to be a d-dimensional vector space over K

#

with basis (1, x, x^2, .., x^(d-1))

#

you can verify that this is a basis

timber bay Jan 24, 2019, 11:28 PM

#

and Q(sqrt(2)) is like Q[x]/x^2-2?

thorny slate Jan 24, 2019, 11:28 PM

#

yeah

#

and the basis is (1, x)

#

so (1, sqrt(2))

timber bay Jan 24, 2019, 11:30 PM

#

okay that makes sense

#

so for B we're supposed to find an element so that when you mod out by it you get a degree 4 vector space?\

thorny slate Jan 24, 2019, 11:34 PM

#

yeah

#

so you want an element in Q(sqrt(2), sqrt(3)) with minimal polynomial of degree 4 over Q

#

that's a tricky way of looking at it though

#

and easier interpretation is to look for an element z such that both sqrt(2) and sqrt(3) are in Q(z)

#

then you know you have an extension of degree 4

#

so here's where the tips we were commenting on come

#

you need an element that has the data of both sqrt(2) and sqrt(3)

timber bay Jan 24, 2019, 11:36 PM

#

what do you mean by has the data?

thorny slate Jan 24, 2019, 11:39 PM

#

that is for you to decide

#

essentially Q(z) = Q(sqrt(2), sqrt(3))

#

as part (c) asks you to show

#

so you can recover both sqrt(2) and sqrt(3) from it

timber bay Jan 24, 2019, 11:40 PM

#

oh okay yeah

#

would something like x^2 - x not work as a way to get both sqrt(2) and sqrt(3)?

thorny slate Jan 24, 2019, 11:43 PM

#

you shouldn't think about it in terms of a polynomial

#

because that would need you to guess the irreducible degree 4 polynomial

#

that said, that polynomial isn't even irreducible

#

it's x(x-1)

timber bay Jan 24, 2019, 11:47 PM

#

arent the elements in E polynomials? shouldnt i be thinking of them as polynomials?

#

unless itd be something like z = sqrt(3)+sqrt(2)

#

or maybe the product?

thorny slate Jan 24, 2019, 11:49 PM

#

kinda

#

I mean

#

they are quotients of polynomials in two variables (x,y)

#

try them out and see what happens

#

it's best to think as them like that

#

to write x = sqrt(2) and y = sqrt(3)

#

and manipulate them as usual

#

for example assuming they are the positive real roots of 2 and 3

timber bay Jan 24, 2019, 11:51 PM

#

so i should do something like set the coeff of a 4th degree polynomial as variables, and solve for them when x and y are solutions?

thorny slate Jan 24, 2019, 11:51 PM

#

no no

#

you're doing it right

#

thinking about sum and product

#

try them out

timber bay Jan 24, 2019, 11:52 PM

#

im not sure how exactly im supposed to maniupulate them

#

sorry if iim just being stupid or something

thorny slate Jan 24, 2019, 11:55 PM

#

for example

#

(sqrt(2)+sqrt(3))(sqrt(2) - sqrt(3)) = ?

#

that's gonna be 2 - 3 = -1

#

that's how you manipulate them

timber bay Jan 24, 2019, 11:59 PM

#

im really struggling to see your point

thorny slate Jan 25, 2019, 12:00 AM

#

we can say sqrt(2)^2 = 2

#

and sqrt(3)^2 = 3

#

that's how you can manipulate them

#

from the polynomials they come from

timber bay Jan 25, 2019, 12:01 AM

#

i dont know what sort of polynomial im looking for

#

or am looking to create

#

i know im doing something stupid

thorny slate Jan 25, 2019, 12:03 AM

#

don't think about the polynomial right now

#

just find an element that gets you both sqrt(2) and sqrt(3)

#

you thought about sum and product

#

try them

#

and see if they work or if they fail

timber bay Jan 25, 2019, 12:03 AM

#

try what

thorny slate Jan 25, 2019, 12:03 AM

#

and get information for that

#

try z = sqrt(2) + sqrt(3)

#

and z = sqrt(2) sqrt(3)

timber bay Jan 25, 2019, 12:03 AM

#

and do what with that lol

thorny slate Jan 25, 2019, 12:03 AM

#

check the degree of Q(z)

#

check if Q(z) contains sqrt(2) and sqrt(3)

timber bay Jan 25, 2019, 12:04 AM

#

oh

#

i need to practice this a lot

#

im really confused.

#

Im having trouble understanding what exactly Q(z) means exactly. I understand its the smallest subfield of R containing z, so containing sqrt(2) and sqrt(3)

#

i dont know how to check its degree or even if it contains those two

#

so far algebra seems to be going over my head a bit

thorny slate Jan 25, 2019, 12:18 AM

#

you have Q(z)

#

so that means you have elements of Q

#

and you can do whatever you want with Z

#

you can multiply it with something else

#

or with itself

#

add it to itself

#

etc

timber bay Jan 25, 2019, 12:18 AM

#

yes

#

ring properties

thorny slate Jan 25, 2019, 12:19 AM

#

so just try to do that

timber bay Jan 25, 2019, 12:19 AM

#

and im trying to find what as a solution? 0?

thorny slate Jan 25, 2019, 12:20 AM

#

you are trying to get sqrt(2)

#

expressing sqrt(2) as a polynomial in sqrt(2) + sqrt(3)

timber bay Jan 25, 2019, 12:30 AM

#

I feel Q(sqrt(2)sqrt(3)) would work

#

Sorry, i just dont know what I'm supposed to do with manipulating z and such

#

unless itd be something like x^2 - 6

#

but that has degree 2

thorny slate Jan 25, 2019, 12:31 AM

#

yeah so that doesn't work

timber bay Jan 25, 2019, 12:31 AM

#

like (sqrt(2)sqrt(3))^2 -6 = 0

thorny slate Jan 25, 2019, 12:31 AM

#

yes

#

so that one has degree 2

#

and doesn't work

#

test the other one

#

try to get sqrt(2) somehow

timber bay Jan 25, 2019, 12:33 AM

#

so when z = sqrt(2)+sqrt(3) does x-sqrt(3) not give us sqrt(2)

#

thats what i go to first

#

i know thats not right

thorny slate Jan 25, 2019, 12:34 AM

#

but you don't have sqrt(3)

#

so you can't do that

timber bay Jan 25, 2019, 12:34 AM

#

okay

#

ill play around some more

#

ok so i got x^3-9x

#

when you plug in (sqrt(2)+sqrt(3)) you get sqrt(2) back

thorny slate Jan 25, 2019, 12:40 AM

#

so you're done

#

cuz sqrt(3) is the difference of both

#

is that correct though

timber bay Jan 25, 2019, 12:41 AM

#

oh i meant to say .5x^3-4.5x

#

and because it has degree 3 it is a 4 dimensional vector space?

thorny slate Jan 25, 2019, 12:42 AM

#

to check that degree you need a polynomial with p(sqrt(2) + sqrt(3)) = 0

#

but it's easier to do this:

#

check that Q(z) = Q(sqrt(2), sqrt(3))

#

one side is obvious

#

the other one is what you have just proved

#

so then it is a 4 dimensional space

#

because the other one is

timber bay Jan 25, 2019, 12:44 AM

#

yeah that makes sense

thorny slate Jan 25, 2019, 12:45 AM

#

anyway you can also find the polynomial explicitly from what you did

#

x^3 / 2 - 9x/2 = sqrt(2)

#

so just square it and subtract 2

#

and there's a polynomial of degree 6

#

it has two extra factors of x which you can remove

#

in other words

#

it's

#

((x^2)/2 - 9/2)^2 - 2

#

a degree 4 irreducible polynomial

timber bay Jan 25, 2019, 12:47 AM

#

oh okay.

#

why does this necessarily contain

#

sqrt(3)

thorny slate Jan 25, 2019, 12:51 AM

#

cuz sqrt(3) = (sqrt(2)+sqrt(3)) - sqrt(2)

timber bay Jan 25, 2019, 12:52 AM

#

ohhh

#

yeah

hot folio Jan 25, 2019, 1:02 AM

#

what was the quick trick to check if a subset B is a subring of A ?

#

for all a,b in B, a x b is in B (x is multiplication)
for all a,b in B, a -b is in B 
B is a subset of A

#

this is what i wrote in my notes

#

but i'm not sure

#

because iirc we have to also check if B is a subgroup

tulip barn Jan 25, 2019, 1:13 AM

#

Isn't the subgroup test just closure?

thorny slate Jan 25, 2019, 1:13 AM

#

the second one tests subgroup

hot folio Jan 25, 2019, 1:17 AM

#

thonker

#

yeah smantha?

#

Majesty what do you mean

#

https://proofwiki.org/wiki/Subring_Test

Subring Test

tulip barn Jan 25, 2019, 1:22 AM

#

The second line is a statement about closure under addition.

#

Which seems sufficient to prove that B is a subgroup.

hot folio Jan 25, 2019, 1:25 AM

#

oh i see, thank you for elaborating

timber bay Jan 25, 2019, 1:32 AM

#

alright. I think it all makes sense now. thanks @thorny slate for all the help and tolerating me being stupid

thorny slate Jan 25, 2019, 1:35 AM

#

yw

timber bay Jan 25, 2019, 1:50 AM

#

is x^4-2 what it's looking for in part c?

📎 unknown.png

thorny slate Jan 25, 2019, 1:52 AM

#

yes

timber bay Jan 25, 2019, 1:52 AM

#

for d itd be x^2-2 yeah?

#

or am I thinking of that wrong

thorny slate Jan 25, 2019, 1:53 AM

#

wait that seems wrong

#

yeah it's wrong

#

it's very similar though

#

do the calculation in paper

timber bay Jan 25, 2019, 1:58 AM

#

hmm

#

is it x^4-2 since x^4-2 +(f(x)) = 0+(f(x))

#

the same thing?

#

@thorny slate why is x^2-2 wrong doesnt that give us 0 ?

thorny slate Jan 25, 2019, 2:19 AM

#

no

#

it isn't x^4 - 2 either because that's not irreducible

timber bay Jan 25, 2019, 2:19 AM

#

oh

#

so its the reduction of that one

#

x^4-2 = 0 but its not irreducible

#

i forgot to include sqrt(2)

#

x+2^(1/4)

oblique river Jan 25, 2019, 2:22 AM

#

2^(1/4) is not in E

timber bay Jan 25, 2019, 2:22 AM

#

x^2+sqrt(2)

oblique river Jan 25, 2019, 2:22 AM

#

close

#

you just have the sign wrong

#

the solution to x^2 + a = 0 is "sqrt(-a)"

timber bay Jan 25, 2019, 2:23 AM

#

how do I know which one to choose?

oblique river Jan 25, 2019, 2:23 AM

#

the square of your element x is sqrt(2)

#

oh wait

#

i'm sorry

#

yeah i'm dumbn

#

good point, so maybe you shouldn't use either of them...

timber bay Jan 25, 2019, 2:24 AM

#

what

#

lol

oblique river Jan 25, 2019, 2:24 AM

#

you should already have a name for "the square of x" in this problem

timber bay Jan 25, 2019, 2:25 AM

#

what do you mean?

oblique river Jan 25, 2019, 2:25 AM

#

how did you do part a

timber bay Jan 25, 2019, 2:25 AM

#

oh so beta

#

yeah

#

beta = x^2

oblique river Jan 25, 2019, 2:25 AM

#

so the minpoly of x over E = Q(beta) is...

timber bay Jan 25, 2019, 2:26 AM

#

beta -2?

oblique river Jan 25, 2019, 2:26 AM

#

uhh

#

beta - 2 is a constant polynomial

timber bay Jan 25, 2019, 2:26 AM

#

not if beta = x^2

oblique river Jan 25, 2019, 2:26 AM

#

so

#

beta is a number

#

(by number i mean an element of Q(beta))

#

it's not a variable

#

your answer needs to be a polynomial with coefficients over Q(beta)

#

such that if you plug in x

#

you get 0

#

and also that is irreducible over Q(beta)

timber bay Jan 25, 2019, 2:27 AM

#

yeah

#

won't it just be a reduction of x^4-2 ?

oblique river Jan 25, 2019, 2:28 AM

#

the issue I think is that the notation is trash

#

we're using x for both the variable and the element of Q[x]/f(x)

#

let's use t as the variable

#

the minpoly of x over Q is

#

t^4 - 2

#

the minpoly of x over E is....

timber bay Jan 25, 2019, 2:29 AM

#

yeah

oblique river Jan 25, 2019, 2:29 AM

#

[this is where you come in :) ]

timber bay Jan 25, 2019, 2:29 AM

#

im thinking

#

lol

#

is it not like beta^2 - 2?

#

does that not equal 0 in this particular extension

oblique river Jan 25, 2019, 2:31 AM

#

that is a constant polynomial

#

there are no variables

#

your answer should be a polynomial, with coefficients in Q(beta), such that when you plug in x you get 0

#

there's nowhere to plug x into "beta^2 - 2"

timber bay Jan 25, 2019, 2:32 AM

#

for beta i was meaning

#

i dont see how you could plug in x into an irreducible polynomial and get 0

#

if your alpha is just x

oblique river Jan 25, 2019, 2:35 AM

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oh I didn't see that we had a name for this element

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I thought we were just calling it x

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we can call it alpha

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i'm not sure what your concern is though

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the minimal polynomial of alpha over Q is t^4 - 2. the minimal polynomial of alpha over Q(beta) is [what]

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your answer should be a polynomial (in the variable t, let's say)

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with coefficients in Q(beta)

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such that w hen you plug in x (or alpha or whatever you wanna call it) you get 0

timber bay Jan 25, 2019, 2:37 AM

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was i wrong when I said x^2 -2

oblique river Jan 25, 2019, 2:37 AM

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yes

timber bay Jan 25, 2019, 2:37 AM

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or is it alpha^2-2

oblique river Jan 25, 2019, 2:37 AM

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that is not a polynomial

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(well, it is a constant polynomial)

timber bay Jan 25, 2019, 2:37 AM

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it is if alpha is a variable

oblique river Jan 25, 2019, 2:37 AM

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alpha^2 is not equal to 2!

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no don't use alpha as a variable

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our only variable from now on is t

timber bay Jan 25, 2019, 2:39 AM

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oh I meant to say t^2-sqrt(2)

oblique river Jan 25, 2019, 2:39 AM

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but we said earlier we weren't gonna call it sqrt(2)

timber bay Jan 25, 2019, 2:39 AM

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did we?

oblique river Jan 25, 2019, 2:39 AM

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because we don't know if alpha^2 = sqrt(2) or -sqrt(2)

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but we do know that alpha^2 is equal to.....

timber bay Jan 25, 2019, 2:39 AM

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so what should I call it?

oblique river Jan 25, 2019, 2:40 AM

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what do you mean? you gave alpha^2 a name in part a

timber bay Jan 25, 2019, 2:40 AM

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i gave x^2 = beta in part a

oblique river Jan 25, 2019, 2:40 AM

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yes

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alpha^2 = beta

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so the min poly of alpha over E = Q(beta)

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is t^2 - beta

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that's it

timber bay Jan 25, 2019, 2:42 AM

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that seems really dumb

oblique river Jan 25, 2019, 2:42 AM

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what?

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that's literally the only answer haha

timber bay Jan 25, 2019, 2:42 AM

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i mean it makes sense and youre right it just seems weird

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like isnt that essentailly x^2-x^2

oblique river Jan 25, 2019, 2:42 AM

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I mean at the end of the day, you're looking for a polynomial such taht when you plug in alpha, you get 0

timber bay Jan 25, 2019, 2:43 AM

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yeah

oblique river Jan 25, 2019, 2:43 AM

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so idk why you're surprised that when you plug in alpha, you get 0

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haha

timber bay Jan 25, 2019, 2:43 AM

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which is why it makes sense

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i was just expecting something more complicated

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i gues

oblique river Jan 25, 2019, 2:43 AM

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algebra doesnt have to be complicated haha

timber bay Jan 25, 2019, 2:43 AM

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i dont think its my expertise

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i need to work on it a lot

oblique river Jan 25, 2019, 2:43 AM

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the point of the problem is just to kinda illustrate that the minpoly depends on the base field

timber bay Jan 25, 2019, 2:43 AM

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its really cool but im bad at it so far

oblique river Jan 25, 2019, 2:44 AM

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and that if you add more elements to your base field (i.e. going from Q to Q(beta)) then your minpoly can get small

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it's just a different way of thinking from whatever else youre probably used ot

timber bay Jan 25, 2019, 2:44 AM

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i guess

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this is my second semester in it

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im still bad at it

oblique river Jan 25, 2019, 2:48 AM

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https://tenor.com/view/dont-stop-dont-give-up-kid-advice-motivation-gif-4496803

Tenor

chilly ocean Jan 25, 2019, 8:48 PM

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anyone willing to help me out with some math?

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its 6th grade math

wise grove Jan 25, 2019, 9:27 PM

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Sure

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@chilly ocean

chilly ocean Jan 25, 2019, 9:27 PM

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thanks

wise grove Jan 25, 2019, 9:27 PM

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If its not abstract algebra, then probably best not to be here

chilly ocean Jan 25, 2019, 9:28 PM

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im not sure which it is

wise grove Jan 25, 2019, 9:28 PM

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What's the problem?

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If you want you can dm me but I understand if you don't want to

chilly ocean Jan 25, 2019, 9:28 PM

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ok here

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to go to the theater you would pay $8:00 for each ticket you purchase. Which answer choice correctly represents the algebraic expression for the cost of t tickets?

stone fulcrum Jan 25, 2019, 9:32 PM

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Definitely not abstract algebra, lel. Abstract is a university course for pure mathies.

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But, it sounds like C = 8t is the correct answer

chilly ocean Jan 25, 2019, 9:34 PM

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ok so where would i ask for help with this?

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im guessing algebra

stone fulcrum Jan 25, 2019, 9:36 PM

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Sure sure

chilly ocean Jan 26, 2019, 4:49 AM

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📎 IMG_1127.JPG

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How do I prove this?

stone fulcrum Jan 26, 2019, 5:39 AM

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@chilly ocean
The inverse of an element "g" is the element you multiply it by to get e.

What's the inverse of ggg... n times?

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(ggg n-times)(g¯¹g¯¹g¯¹ n-times) = e
Because each g cancels out with a g¯¹

knotty lagoon Jan 26, 2019, 6:12 AM

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thonker

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Use induction

chilly ocean Jan 26, 2019, 7:25 AM

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ight thanks man

runic bear Jan 26, 2019, 9:09 AM

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hello guys, if an augmented matrix consists of two identical rows ,would that mean it can't be in row-echelon form since the pivot column would consists of two pivot points (since there's two identical rows)

white turret Jan 26, 2019, 1:31 PM

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@runic bear this belongs in #prealg-and-algebra and the answer is yes: if u got 2 non-zero identital rows, it cant be in row-echelon form

runic bear Jan 26, 2019, 4:04 PM

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my apologies and thank you so much! @white turret

wise grove Jan 26, 2019, 5:12 PM

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Thats technically a linear algebra question

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But could also be perceived as an algebra 2 question

somber jolt Jan 29, 2019, 4:48 PM

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hello. i know that S(E,o) is a group (permutations of E with composition law)
but i don’t understand why ? can someone explain it to me ? thank : )

viral bear Jan 29, 2019, 4:49 PM

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Why don't you try to prove it, and we can help if/where you get stuck?

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You remember how to prove something is a group?

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We also have latex in this server btw $\bR$

somber jolt Jan 29, 2019, 4:51 PM

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i think to prove that * is associative, E has a neutral element, every element of E has a reverse

cloud walrusBOT Jan 29, 2019, 4:51 PM

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ixsetf:

📎 90266169730146304.png

somber jolt Jan 29, 2019, 4:51 PM

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yup, do you want me to write it in LaTex ?

viral bear Jan 29, 2019, 4:51 PM

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If you know how yes that would be best

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Also just as a note

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What you are calling a neutral element is the identity

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And what you are calling a reverse is called an inverse

somber jolt Jan 29, 2019, 4:53 PM

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You mean the application Id right ?
thanks !

viral bear Jan 29, 2019, 4:53 PM

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Application Id?

somber jolt Jan 29, 2019, 4:54 PM

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identity function

viral bear Jan 29, 2019, 4:54 PM

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In this case you are right

somber jolt Jan 29, 2019, 4:54 PM

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alright. i’ll try to prove it when i’m home, then write in latex if needed

viral bear Jan 29, 2019, 4:55 PM

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Ok cool

chilly ocean Jan 29, 2019, 7:25 PM

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hey guys, anyone who could help me with linear mapping? i have a few questions regarding the subject

somber bramble Jan 29, 2019, 7:46 PM

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JustAsk

chilly ocean Jan 29, 2019, 7:48 PM

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so i don't quite understand how the vectors are put into linear map's matrix, let's say i have a 2 dimensional space, and i want a linear map that rotates my vector 90 degrees

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A(i)=j

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A(j)=i

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and i put that into a matrix, why am i putting it into columns and not into rows?

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erm i made a mistake here

somber bramble Jan 29, 2019, 7:49 PM

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alright, so first of all, you need to pick some basis. that is, you need to just pick out two (because two-dimensional) vectors which you will represent everything in terms of. those would be your i and j here

chilly ocean Jan 29, 2019, 7:49 PM

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A(j)=-i

somber bramble Jan 29, 2019, 7:49 PM

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so by definition then i = (1,0) and j = (0,1)

chilly ocean Jan 29, 2019, 7:49 PM

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yes

somber bramble Jan 29, 2019, 7:49 PM

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right? (those are column vectors)

chilly ocean Jan 29, 2019, 7:49 PM

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yes

somber bramble Jan 29, 2019, 7:50 PM

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now you want to find a matrix A so that Ai = j and Aj = -i

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so you need to think, what happens actually when you do A*(1,0)

chilly ocean Jan 29, 2019, 7:50 PM

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aye, i get that

somber bramble Jan 29, 2019, 7:51 PM

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$$\begin{bmatrix} a & b \ c & d\end{bmatrix} \begin{pmatrix} 1 \ 0\end{pmatrix} = \begin{pmatrix}a \ c\end{pmatrix}$$

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fuck

chilly ocean Jan 29, 2019, 7:51 PM

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😛

cloud walrusBOT Jan 29, 2019, 7:51 PM

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Sascha Baer:

📎 131832027082260480.png

somber bramble Jan 29, 2019, 7:51 PM

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there

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so this tells you what vector (1,0) will be mapped to

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now you can simply fill in the values for a and c as you desire

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repeat the same with (0,1) and you’ll see that you’ll get out (b,d), and so you can fill in those as you desire

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and then you’ll have your matrix

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so the reason why we fill in the columns is: because matrix multiplication then gives us the right thing

chilly ocean Jan 29, 2019, 7:53 PM

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hmm ok

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but does the actual column of matrix

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have to do something with values next to i and j

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i understand what happens when u multiply it

somber bramble Jan 29, 2019, 7:54 PM

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wdym values next to i and j?

chilly ocean Jan 29, 2019, 7:54 PM

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aye

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erm

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like

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is the matrix somehow directly connected in a way in which it affects the vector

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for example are these a and b representing i?

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and c and d represent j?

somber bramble Jan 29, 2019, 7:55 PM

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i = (1,0) by your definition

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the first column of the matrix tells you where i gets mapped to

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and the second column where j gets mapped to

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as we just discovered

chilly ocean Jan 29, 2019, 7:55 PM

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ye

somber bramble Jan 29, 2019, 7:55 PM

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and if you have, for example a new vector v = i+j

chilly ocean Jan 29, 2019, 7:55 PM

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ye

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does i still get mapped to j?

somber bramble Jan 29, 2019, 7:56 PM

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then v will get mapped to the sums of what i gets mapped to + what j gets mapped to

chilly ocean Jan 29, 2019, 7:56 PM

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ahhh

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ok 😄

somber bramble Jan 29, 2019, 7:56 PM

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because linearity:

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T(v+w) = T(v) + T(w) for linear maps

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and matrices represent linear maps

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so in particular, if v is, say 3i + 4j, then T(3i + 4j) = 3T(i) + 4T(j) = 3j - 4i

chilly ocean Jan 29, 2019, 7:57 PM

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ok wait

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so we said i+j

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i gets mapped to j

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and j gets mapped to -i?

somber bramble Jan 29, 2019, 7:57 PM

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yea, that’s what you wanted it to do

chilly ocean Jan 29, 2019, 7:57 PM

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yes

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so i mean i could solve it this way

somber bramble Jan 29, 2019, 7:57 PM

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T(i+j) = T(i) + T(j) = j - i

chilly ocean Jan 29, 2019, 7:57 PM

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without matrix multiplication

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i get that it's hard

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with n dimensional spaces

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but ok, i got an understanding now 😛

somber bramble Jan 29, 2019, 7:58 PM

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matrix multiplication is tedious, linear maps are very nice

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I avoid matrices when I can

chilly ocean Jan 29, 2019, 7:58 PM

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hmm ok, i have another question

somber bramble Jan 29, 2019, 7:58 PM

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if I have a good understanding of what the linear map does, using a matrix can often be more annoying than anything

chilly ocean Jan 29, 2019, 7:58 PM

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so linear map is a function basically

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from X->?

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Y*

somber bramble Jan 29, 2019, 7:58 PM

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in the most general case, between two vector spaces over the same field

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T: V→W

chilly ocean Jan 29, 2019, 7:59 PM

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same field

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what does that mean 😛

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i'm learning all of this in a different language so it's hard hahahha

somber bramble Jan 29, 2019, 7:59 PM

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well, in a vector space you have scalar multiplication, right?

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those scalars have to be from some set, e.g. real numbers, complex numbers, 𝔽₂…

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you can’t mix those

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if V has real numbers, then W must too

chilly ocean Jan 29, 2019, 8:00 PM

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ah aight

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i get it 😛

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so i wanted to ask, is base transition matrix also a map?

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base/basis how do u call it 😛

somber bramble Jan 29, 2019, 8:01 PM

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yea, change of basis is also a linear map/function (map and function really just mean the same thing idk why I always call them maps)

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it’s kind of weird to think about it tho

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also, in my opinion, a matrix is not a function per se

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it just represents one

chilly ocean Jan 29, 2019, 8:02 PM

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so when im making a map matrix i just want to put into columns where i want vectors in the basis i am in to be mapped to?

somber bramble Jan 29, 2019, 8:02 PM

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yes

chilly ocean Jan 29, 2019, 8:03 PM

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aight

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erm btw is there a reason why matrix multiplication is the way it is?

somber bramble Jan 29, 2019, 8:03 PM

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“because it works out well that way”

chilly ocean Jan 29, 2019, 8:03 PM

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xD

somber bramble Jan 29, 2019, 8:03 PM

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I agree that it really just seems to fall from the sky

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and be weird

chilly ocean Jan 29, 2019, 8:04 PM

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ye, it's all weird rn, coming over from high school math's to this abstract stuff is kinda hard to comprehend

somber bramble Jan 29, 2019, 8:04 PM

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it’s just that it’s the only way to nicely represent linear functions

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cause the way it is defined is more or less by the rules of linear functions, you know?

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it’s best to undersand matrix multiplication as a generalization of matrix-vector multiplication

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convince yourself that matrix multiplication is exactly the thing you need for linearity. that is, if A is a matrix and v, w are vectors, α is a scalar, then you must have:
A(v + αw) = Av + αAw

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(that’s the definition of linear I always work with, you could also define it in two steps, as A(v+w) = Av + Aw and A(αv) = αAv)

chilly ocean Jan 29, 2019, 8:06 PM

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aye

somber bramble Jan 29, 2019, 8:06 PM

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convince yourself that these are the same things

chilly ocean Jan 29, 2019, 8:06 PM

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aight, thank you so much for the help 😃

stone fulcrum Jan 29, 2019, 8:36 PM

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@chilly ocean
One way to see a matrix is as an operator for vectors. Vectors go in, vectors go out.

Matrix multiplication ensures that:
(xA)B = x(AB)
That is, if you pass a vector through a matrix, then another matrix, you could have instead multiplied the two matricies together, THEN passed the vector through the product.

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Multiplying two matricies retains what they each do to a vector.

chilly ocean Jan 29, 2019, 8:40 PM

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@stone fulcrum thanks 😃

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am i allowed to post images here related to subject? i have a few more questions 😛

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@somber bramble hey are u still here i have a question regarding the basis change

somber bramble Jan 29, 2019, 8:52 PM

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sry, going to bed; Also change of basis is a topic where I always manage to confuse myself if im not fully awakw

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so i'd rather not say sth wrong

chilly ocean Jan 29, 2019, 8:53 PM

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ahh aight 😃 thanks anyways 😃 good night