groups-rings-fields

[final pasture]

(4)

wait

i'm

[wild sapphire]

are you not talking about x^2 + 1

[final pasture]

a bit tired

[wild sapphire]

instead

[final pasture]

yes

[wild sapphire]

yeah

[final pasture]

sorry lol

so X² + 1 xD

no roots in Z

but a root (which is 4) in Z/5Z

[wild sapphire]

im considering the function (2x+3)(x^2+1) This doesn't seem to have roots in Z5 or even Z

wait,

but with what you gave me, i guess it should?

[final pasture]

well 4 is a root

since 4**2+1 = 0 [5]

[wild sapphire]

yeah

hmm. I guess I'm relying too much on graphical intuition

[final pasture]

although

Z/5Z is definitely not algebraically closed

there's definitely polynomial in Z/5Z[X] with no roots in Z/5Z

take for example P = x(x-1)(x-2)(x-3)(x-4) + 1

[wild sapphire]

ah yeah

[final pasture]

P(x) = 1 for any x € Z/5Z

[wild sapphire]

interesting

[final pasture]

(note that this method generalizes, it works to show that any finite ring can't be algebraically closed )

[wild sapphire]

so, how could I go about finding what makes a root within Z5? like would 4 be the only root of x^2+1 or just one of them ?

[final pasture]

Mmh

I remember a few months ago a friend of mine explained me something about that

give me a minute I'll try to find what he explained me exactly

[wild sapphire]

Like i guess in this case I can plug in all 5 values

to my functions

[final pasture]

like if I remember correctly it was that the usual methods for solving quadratic equations over C worked too over Z/pZ for p a prime

except that the square roots aren't always defined, and some details like that

[wild sapphire]

yeah, like the quadratic formula. But the quadratic formula on x^2+1 would give me complex roots and i don't know how from there it would indicate that 4 is a root yknow

[final pasture]

no it wouldn't if you use it over Z/5Z (I think)

So like

let me try

x^2 + 1
Delta = -4 = 1 [5]
sqrt(1) = 1 [5]
Solutions should be pm sqrt(Delta)/2, so -sqrt(1)/2 = -1/2 and sqrt(1)/2 = 1/2, the inverse of 2 mod 5 is 3, so solutions are 3 and 3*-1 = 2 [5].
Both are solutions

not sure if that's clear

[wild sapphire]

so you reach a conclusion of 3 and 2 being roots?

[final pasture]

yes

[wild sapphire]

ah hmm

it think i get it

one question

you reached 1/2, but then said that the inverse of 2 mod 5 is 3. I'm getting confused with additive inverses and multiplicative inverses there

[final pasture]

I'm speaking about multiplicative inverse

so basically to solve the equation $ax^2 + bx + c$ over $\bZ/n\bZ$, you should try the values given by $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.\
You should interpret the division by $2a$ as a multiplication by the inverse of $2a$ mod $n$ (if it exists, obvsly, but if $n$ is prime and $a \neq p$ and $p \neq 2$, you're good)and the $\sqrt{b^2 - 4ac}$ as huh.. something that works squared gives you $b^2 - 4ac$.\
Tbf I'm not sure about the details (mostly about the square root, since there's not always a square root what happens in this cae), you should probably try to do a proof of this formula over $\bZ/p\bZ$ for some prime $p$ yourself and see how properly that works, and then try to see what generalizes over $\bZ/n\bZ$ for any integer $n$

[lunar coyote]

when they say every nonzero element is a unit does it mean it has an inverse?

[cloud walrus]

Shika-Blyat

[wild sapphire]

@lunar coyote non-zero does not imply it has an inverse in general

oops unit does tho

yeah that's what unit means

so either it has an inverse or it divides zero

[final pasture]

now when something isn't defined (like when the multiplicative inverse doesn't exist or there's no square root), I'm not sure about what happens. Maybe there's no root, maybe there can be but we can't find it like that, idk

[wild sapphire]

i think i understand @final pasture

[final pasture]

not sure if that makes things clearer though

ok great

[wild sapphire]

I have one more question related to groups ive been stuck on

[final pasture]

go ahead, ask

[wild sapphire]

are there any groups such that taking the power m of all elements doesnt form a subgroup.

So for $(G, \cdot), n \in \mathbf{N}, G_n = {g^n : g \in G}$

[cloud walrus]

darkninja175

[wild sapphire]

maybe I don't know enough about group structures but I can't think of one

[final pasture]

I believe it is always a subgroup and I don't think it'd be hard to show it is

So take a group (G, .)

and some integer n € N

[wild sapphire]

it's always a subgroup if G is abelian, I've seen that before

[final pasture]

oh wait you're right

i'm assuming commutativity

So yeah, if it's abelian it is a subgroup

and if it's not abelian ig it's not, mmh

[wild sapphire]

i've been trying to think of what group would hold this property of violating the subgroup axioms, but i only really have like well behaved groups in my head that dont seem to do it

[final pasture]

So I didn't check but I think this would fail with S_3

for n = 2, 3 or 5 (one of those, not sure which one, just intuition speaking )

[wild sapphire]

I feel like I checked S_3 but i can give it another quick check

[final pasture]

take 2 elements of S_3 that doesn't commute

And something should fail with them

[wild sapphire]

So like in D3 s doesn't commute with r. 🤔 you're saying i should focus on those

for 2 we get {e, r, r^2}
for 3 we get {e, s, ps, p^2s}

oh but for D3 and the power of 3 that doesnt seem like a nice group

I don't think {e, s, ps, p^2s} is a group it doesn't seem closed

[final pasture]

[hidden haven]

Yes

[wild sapphire]

it cant be closed because ps * s = p, am i thinking correctly or is this just gibberish

[hidden haven]

Take free group on {a,b}

aa • bb = aabb is not a square

[final pasture]

No you're right

[hidden haven]

So not closed under the operation

[final pasture]

And moldilocks' example is nice too

[hidden haven]

[final pasture]

what's k ?

[hidden haven]

And >? You won't have order in an arbitrary ring

[final pasture]

[lunar coyote]

whats order?

[hidden haven]

In a ring, you can't talk about elements being greater than other elements

In Z, R etc you can, but not in general, like how would you define order on polynomials?

[lunar coyote]

through degree?

[hidden haven]

That's only a partial order, and that kind of thing will again only be possible in Euclidean domains

[final pasture]

that's not even a partial order

[hidden haven]

Even for Z/nZ, you can't talk about order because it loops back

[final pasture]

the "order" that the degree of polynomials gives doesn't respect antisymetry

you don't have deg(P) <= deg(Q) and deg(Q) <= deg(P) implies P = Q

[hidden haven]

Right, pre order

[drowsy quest]

cant you order polynomials by degree and dictionary order for same degree

or no

[final pasture]

Also you can talk about order in Z/nZ, it's just that you can't talk about an order respecting the ring structure of Z/nZ

[hidden haven]

^ yeah this

Same for the dictionary ordering on polynomials

[drowsy quest]

gotcha

[final pasture]

(but atleast dict ordering gives a total order yeah )

[drowsy quest]

can someone confirm that this question im given makes no sense

part b

not a field bc the mult still doesnt commute

[hidden haven]

Yeah it should be division ring

Non commutative field

[final pasture]

^

[drowsy quest]

ty i had to make sure im not going insane lol

[lunar coyote]

is it concerning that i can barely do any questions from my abstract algebra book

[hidden haven]

Are you taking a course or self studying?

[final pasture]

Also a question I meant to ask since a few days Moldilocks:
What does 🍎 mean ?

[hidden haven]

eat 🔫

[final pasture]

also which book, redd ?

[lunar coyote]

i'm taking a course but is taught extremely badly

contempary abstract algebra

[drowsy quest]

if its online that probably doesnt help

[lunar coyote]

all online

[old lava]

gallian?

[lunar coyote]

yep

[old lava]

change asap

gallian 🤮

[lunar coyote]

😢

[final pasture]

Alright alright, don't shoot 🤚 😄 ✋

[drowsy quest]

is that the duluth reu guy

[hidden haven]

Had this with my group thy course it is somewhat concerning because it will affect your grades and such but in the long run it will probably turn out fine, because when you find a better course that requires group thy you'll have a lot more motivation and background to do it on your own

[lunar coyote]

was recommend "peter cameron introduction to algebra" by my lecturer but it's pretty bad imo

[hidden haven]

Like I re did all group thy on my own during my gal thy course in just a few days

[lunar coyote]

also should i change my book? lol

sounds like it's bad or something

[final pasture]

If you're struggling with the current one, yeah probably

[old lava]

use d&f or artin or like anything else really

than gallian

[final pasture]

idk if d&f is a good recommendation for a first course

[hidden haven]

Every single student from my college hates artin

[old lava]

d&f is literally one of the easiest algebra books though

[hidden haven]

So probably don't use it

[final pasture]

and artin is well.. particular, either you love or you hate it

[lunar coyote]

i'm not actually studying abstract algebra specifically but my course has some content on ring theory

[final pasture]

What's your course about ?

[old lava]

d&f requires minimal mathematical maturity

so does gallian, but gallian just doesn't cover half the important topics

and also presents some things way too late/in a bad way

[drowsy quest]

is it a cryptography course or something

[final pasture]

ok so that's an elementary number theory course redd, right ?

[lunar coyote]

idk

the course name is just called algebra

[final pasture]

Actually

[old lava]

looks like a rings course really

with some elementary number theory topics attached

that easily follow from intro ring theory

[lunar coyote]

but taught extremely poorly

lecturer finished the course and i still don't understand whats an ideal

[old lava]

that's not ideal

[final pasture]

[old lava]

pun actually not intended

[final pasture]

I don't believe you

[old lava]

I realized it right when I pressed enter

[final pasture]

[lunar coyote]

ye

so long story short i can barely do the questions on this book

kek

should i continue

and keep fighting or change books

[old lava]

idk, gallian is kinda poor

imo

d&f on the other hand is more than enough for your course

and has great exercises

but is incredibly dry

which puts many people off

[final pasture]

The thing is your course is covering two distinct (although related) topics, so I feel like recommending you an abstract algebra book would be too much since you're only looking for basic ring theory but I also feel that recommending you an elementary book theory wouldn't help since you also need some stuff about the said basic ring theory and idk any elementary number theory book that speaks about ring early enough

I'd say you should forget gallian, atleast

but idk what to recommend instead

I think d&f would be too much + may be too hard + as F[x] said, it's D R Y, which makes it hard to read imo

[old lava]

d&f would be too much for your course

but your course is kinda all over the place

d&f alone does cover all of that courses's content

[lunar coyote]

is abstract algebra normally a 3rd year course?

[old lava]

albeit in a different order

yes

[lunar coyote]

i'm 2nd year lul

[final pasture]

(and that's a problem, in this case )

[chilly ocean]

Yeah, I am not even sure what it means, things like "graduate level course", it seems kind of arbitrary to me

[old lava]

it's just a bit of a strange course

basic abstract algebra can unironically be taught to a grade 9 student who's willing to put the effort into learning it

[final pasture]

(in France, elementary abstract algebra is often covered in the first months of the first year )

[old lava]

in NA, afaik, algebra is typically 3rd year

not including lin alg

[final pasture]

it is ??

Are you implying some people think it is not ?

[strong valve]

abse

based

[final pasture]

oh, right, sorry, i misunderstood

[hidden haven]

Is abse how they spell it in france

[final pasture]

xD

[strong valve]

abse based abse

[old lava]

the biggest barrier to intro abstract algebra isn't really the prerequisite knowledge to it (which is barely anything), it's mostly just the mathematical maturity

[final pasture]

(friends of mine translated "based" to "soclé" in French, which is quite the funny translation when you understand it )

[chilly ocean]

I think the biggest barrier is that they teach you group theory without motivation

[old lava]

I mean it's just kinda standard that algebra is typically taught in 3rd year (in NA at least)

[final pasture]

this this this

[old lava]

which is why it's called a 3rd year course

doesn't mean non-3rd years can't take it

idk, I like group theory without any motivation

it's a pretty theory without anything extra

needed

I feel like it's just too abstract for a lot of people to handle

[final pasture]

I actually like it too, that's how I started maths, but it's so much better (and helps a lot more people that struggles with it) with motivation

[old lava]

before they're more familiar with more proof based math

[final pasture]

Also, this proof-based vs non proof-based thing in NA is definitely shitty weird

[old lava]

"typically"

[chilly ocean]

Yeah, I suspect that at most good schools, students take it 1st or 2nd year

[old lava]

I mean I can't help that hs students don't know proof based math

and spent their lives grinding computations

[final pasture]

hs ?

[old lava]

high school

[final pasture]

oh right

well

the thing is

why is there still course that aren't "proof based" after high school

[old lava]

because 90%+ of the uni population doesn't need proof based math?

they need to know what a derivative is, what an integral is, and basic computational lin alg

at max

[final pasture]

I mean in a pure maths cursus

[chilly ocean]

Is mv calc proof based in france? But I would hate to eg proof stokes theorem

[old lava]

algebra courses have a ton of people that aren't in pure math

[final pasture]

mv ? multivar calc ?

me slim ?

[old lava]

me?

I was talking to shika

[viral jolt]

didn't know I was going to see you there shika

🤣 h

[final pasture]

oh hi lol

[viral jolt]

Heyy

How are you ? (in english of course)

[chilly ocean]

Ah, is this english class?

[viral jolt]

We are on the same maths/physics french server

[chilly ocean]

"Hi, I am doing well. And you?"

[viral jolt]

I had no clue to find him here

[final pasture]

Yeah, although in more advanced course (like multivar calc, which is partially covered in 2nd year usually, and "finalized" 3rd year course), proofs of this kind of annoying theorems may be covered in class but are not emphasized if they're not of particular interest

But yeah, mv calc is proof based here, so in pure maths cursus you'll usually have proofs of big theorems, with epsilons and deltas everywhere

In engineering schools I believe it's not always the case

[chilly ocean]

[viral jolt]

Depending what kind of stuff you are doing in engineering schools

[final pasture]

Exactly what 8da said, so "Hi, I am doing well. And you ?"

Yeah, as h. says .
In "prestigious" engineering schools, it's likely that most of the stuff will be covered with great details (atleast when the details are worth being covered, which may not be the case of important theorems with annoying but not particularly insightful proof ), but in "less prestigous" engineering schools, there's pretty much nothing that is "proof-based"

[viral jolt]

doing alright!^^ I'm definitely in the wrong section (abstract algrebra)

[final pasture]

I feel like I'm repeating myself

[final pasture]

Quotient the space of the channels by the relation #groups-rings-fields ~ #chill, and you won't be wrong anymore

(appreciate how I'm in the right channel for my joke )

Also I love this emoji

[hidden haven]

[final pasture]

I wasn't expecting too see you here either. You were going to ask something or you were just lurking ?

[viral jolt]

lurking obviously, also maybe share a video I saw about Grothendieck but it's in french anyway :))

[final pasture]

No youtube automatic translation ?

(and share it anyway, I'm interested )

's fanboy here

[viral jolt]

https://www.youtube.com/watch?v=pOv-ygSynRI&t=953s

[final pasture]

thanks

[viral jolt]

You ahve the Bourbaki following up in trends after that one

[drowsy quest]

hey that guy wrote my textbook!

[final pasture]

Serre, I guess ?

He wrote so many great books lol

[viral jolt]

I was doing some research on Galois, and his acceptance at X

[final pasture]

He didn't get accept at X, did he ?

Only the ENS accepted him, right ?

[final pasture]

[viral jolt]

Yes he got rejected twice

so he did join the ENS

but after a bit of time, he also got rejected too from there

he went to prison at some point

[drowsy quest]

what does join the ENS mean

ENS is a school in france right

[final pasture]

yeah

actually now it's 4 schools

[drowsy quest]

actually my math professor right now is from france

i think he is from cachan??

[final pasture]

that's one of the 4 ENS, yeah

(Cachan is now called Paris-Saclay)

[drowsy quest]

i see

you are in france?

ive heard of ecole polytechnique too its a good school i suppose

[final pasture]

Yeah, but I'm not in any ENS lol, I'm still in high school

[viral jolt]

It's considered as the best math school in the world

ENS Paris Saclay

[final pasture]

[drowsy quest]

high school

LOL

wow

[final pasture]

Ulm > PS h.

Yeah, polytechnique and the ENS are considered the best schools in France (and also the hardest to join )

[viral jolt]

(he is still in highschool but can talk with doctor in maths about things i have no clue about)

[drowsy quest]

yeah seems impressive 🙂

im a statistics student cause i wasnt good enough at math

[final pasture]

Are you thinking about the category theory thingy with SEV ?

[drowsy quest]

well im a math undergrad but next year im starting grad school in statistics

[nova plank]

Shika in high school. I'm shook.

[viral jolt]

yes

[final pasture]

I also didn't know what I was talking about lol, I'm a total noob in CT

[nova plank]

How can my nemesis be a high schooler?

[final pasture]

[viral jolt]

Concours Général --> IOM next year Shika

think about it

:))

[final pasture]

How does getting crushed by an high schooler feel ? @nova plank

()

[final pasture]

(in french this time: la fleeeeeemme )

[nova plank]

I'm going to sleep...forever

[final pasture]

I don't even want to join a preparatory class

[drowsy quest]

does france have entrance exams for university

[final pasture]

So I don't really care about IOM or the CG, since that only helps getting into such schools, fac is open for everyone and that's probably a better place for me

[viral jolt]

-_-, ruining a potential

[final pasture]

Didn't say I don't want to join an ENS though

[viral jolt]

AH

What did you think about X/ENS for MP this week ?

he is going to say "easy" I swear

[final pasture]

I just don't feel like I'd be able to give the better of myself in preparatory class though

[viral jolt]

Well I wasn't confident either before joining them

[final pasture]

depends on what you call univ ig

[viral jolt]

And the fact I had no clue about actual sciences before doing them didn't help choosing

but it really helped me gain interest and not be lazy anymore

[drowsy quest]

can you think of SU(2) as rotating quaternions the same way that e^itheta rotates complex numbers

[final pasture]

(maths A was really full of classics. Except that idk, I'm too bad in analysis/proba to have an actual idea of what the subject were about or how hard they were )

[viral jolt]

I see, well yes, I was a bit disappointed about Maths A tbh

But the computer sciences last exam was nice

[final pasture]

Well they're both rotating stuff, so yeah ?

[viscid pewter]

wikipedia says yes, i think

which is nice

[final pasture]

(Maybe you're asking something more specific, it this is the case I'm missing it)

[final pasture]

which one ?

[viral jolt]

https://www.cpge-paradise.com/Concours2021/MathInfo.pdf

[final pasture]

there was a few computer science exam recently, centrale's exam and also info A iirc

oh and math info yeah

[drowsy quest]

oooo beautiful

[final pasture]

xDDD

at the end of the subject:

[viral jolt]

8th page

better ❤️

[final pasture]

oh wait yeah it's not even the end xD

LLG is everywhere

[viral jolt]

😘

[drowsy quest]

whats llg

[viral jolt]

Louis Le Grand

the best highschool/preparatory classes in maths

[final pasture]

A preparatory class famous for being the most prestigious one here

and also the one with the best results yeah

[drowsy quest]

is this an entrance exam??

[final pasture]

no

so basically

[viral jolt]

it's one of the many exams you take

[final pasture]

in France

you can go to the univ

but you can also do 2 year of preparatory class

to prepare entrance exams

for some prestigious schools

like the ENS

polytechnique

and other (mostly engineering) schools

this is one of the entrance exams

[viral jolt]

(Centrale)

[drowsy quest]

i see

[viral jolt]

:))

[final pasture]

for the ENS, more specifically

[viral jolt]

(CENTRALE)

[drowsy quest]

so what is this exam for

[final pasture]

Mmh ?

[final pasture]

oh and yeah, centrale exists too

[viral jolt]

it's part of the exams to enter Polytechnique / ENS

[final pasture]

and is one of the worst engineering school you can get ^^

[viral jolt]

considered as one of the hardest ones

[drowsy quest]

so you would take it after you do 2 year of preparatory school in order to enter ENS?

[viral jolt]

to take to enter into engineering schools

Exactly

[final pasture]

(I'm joking, obviously, it's one of the best engineering school here)

[viral jolt]

sean

[drowsy quest]

ah i see

[viral jolt]

I'm at Centrale, that's why he is trolling me

[drowsy quest]

i cant read french but it definitely doesnt look easy

[final pasture]

(^ )

I can read french and it definitely is not easy

[viral jolt]

https://www.cpge-paradise.com/Concours2021/MathA.pdf

that's the maths A subject we were talkign about

which is considered classic as Shika said

[final pasture]

to give some context

in preparatory classes

exercises are extremely important

[drowsy quest]

i only partially understand the questions but i think most senior math students at my school would not pass this exam

[final pasture]

so good students, those who aim to enter at the most prestigious schools, like ENS, do a lot of exercises, and when I mean a lot it's really a lot

And they often use the same (online website or books) resources for that

[drowsy quest]

it makes sense

most of my professors are from europe and asia and they talk about similar things

[viral jolt]

It's a week full of 8 hours of exam everyday, you have 4 actual math tests

A, B, C and D obv

[final pasture]

so we call "classics" exercises that appears quite often in those resources

[viral jolt]

so you are multidisciplinary in that sense

[final pasture]

(and some people call something classic if they know it (just 'cause they want to flex, tbh ))

[drowsy quest]

i see

[viral jolt]

Like you Shika 😏

[drowsy quest]

so how old are people when they finish preparatory school and start university?

[final pasture]

obviously, my sole goal is to flex on you guys

[final pasture]

20 usually

but it's customary if you fail the first time

to do a third year of preparatory class

[drowsy quest]

i see

[final pasture]

(that happens a lot, I'd say something close to half of the people failing do this 3rd year)

by failing

I mean just not having a school you want

you can always go to classic univ

or to a school that accepted you but that is not as good as you wanted

[drowsy quest]

is university paris 7 good

[final pasture]

hmm depends

[drowsy quest]

my professor studied at ENS cachan and then did phd at university paris 7

i just checked his cv :p

[viral jolt]

I was 16 when I entered preparatory classes, and someone was 15

so it really depends, but yeah most people are 18 when they start their two years of preparatory classes

[final pasture]

first 3 years => the overall level of the student is hmm.. how to put: garbage, like every univ in France ()
after that => It's a good university yeah

(Idk if h. will agree with me when I say that every univ in France is garbage in the first 3 years )

[drowsy quest]

i know what you mean

[viral jolt]

I Agree

One exception : Dauphine

Sean if you are interested : https://www.sieben.fr/2014/01/02/prepa/

[final pasture]

(Hmm, yeah Dauphine isn't that bad, still nowhere as near of a "bad" preparatory class though )

[viral jolt]

that's an article about what is french prenpa

it's really unique

and has its own vocabulary in a sense

"they won’t all go to the Ecole Polytechnique. The renowned engineering schools are Centrale Paris, Supélec" Facts

😭

[final pasture]

lol

[drowsy quest]

thanks, i'll read it

[final pasture]

Also I don't like the prepa system personally, but mostly because I don't think I'd like it lol

[viral jolt]

You can't know before you try

[final pasture]

Well I'm not planning on trying

although

Ok

I'm planning on trying

but only one prepa

[chilly ocean]

Wait, you are in france and you are not in prepa?

[final pasture]

I'm in high school

[final pasture]

Because I made a promise to a friend, that I'd try join his ex-prepa @viral jolt

[viral jolt]

And I finished mine !

[chilly ocean]

But so you do not want to go to ENS?

[viral jolt]

(fr : Laquelle ?)

[final pasture]

I don't, but I'm planning on doing it through the univ

(Thiers, une prépa marseillaise +/- random)

[viral jolt]

Not random at all

[final pasture]

(enfin elle a quand même été la prépa du major de l'X y'a 2 ou 3 ans )

[final pasture]

Wait

Idk how to write

I meant I do @chilly ocean*

[drowsy quest]

"a student is called a 3/2 if he integrates the Ecole Polytechnique between his first and second year of preparatory class since the integral of x from 1 to 2 is 3/2"

lmfao

[final pasture]

yeah xD

and so if you do this 3rd year

you're called 5/2

[lyric delta]

hey sorry quick question, where do i post general relativity questions

[final pasture]

probably on the physics server

[final pasture]

But I don't have a link sorry

[viral jolt]

and fields winner

[final pasture]

Ah oui y'a Connes qui est passé par là aussi

Connes-sensei

[viral jolt]

Connyyyyyy teach me the ways 😆

[drowsy quest]

so when students finish prepa to start ENS or something, they have a good understanding of algebra, analysis,topology,etc already?

[final pasture]

mmh

highly depends on the student lol

linear algebra, analysis, topology on real normed vector space are covered in prepa

in some details

abstract algebra is covered, a bit

but point-set topology isn't at all

multivar calculus isn't covered in great details either

metric spaces aren't either (even though it's customary to speak about them a bit atleast)

[final pasture]

but even stuff like quotient structures isn't supposed to be covered

some "good" preparatory classes do it, but not enough so that we can say it is "properly covered"

forget stuff like group action or repr theory

As I said preparatory classes have a great culture of the exercises

and more particularly

of hard exercises

[drowsy quest]

got it

[final pasture]

of exercises that are particularly technical

[lyric delta]

when did u guys really start to "enjoy" maths like im in 3rd year undergrad and still dont rly enjoy it

[drowsy quest]

sometimes i like it and sometimes i hate it :p

when i studied measure theory is when i really liked math

cause i liked the subject a lot

algebra is more boring and nonintuitive for me

[final pasture]

So the subjects covered aren't particularly wide and aren't covered particularly deeply, but someone who finished prepa has already seen a fair bit of what we could call "completely" elementary mathematics, yeah, even though there's holes in what should be covered to say that @drowsy quest

[lyric delta]

was there like a breaking point like oh wow maths is acctually enjoyable

[final pasture]

I enjoy maths when I learn something particularly beautiful

and that happens often enough to keep me interested over the time

[drowsy quest]

i dont really know lol. i like to learn math but i dont know if i find it enjoyable

maybe this is why im quitting math for statistics

but the school im going to go to is known for being very theoretical for statistics

half the faculty are almost like probabilists

[final pasture]

are you saying this like it's a bad thing or are you glad it's the case ?

[drowsy quest]

im glad

probability is my favorite subject

tbh, half the reason im doing stats is bc i was accepted into a much better statistics department vs math department

[final pasture]

Also in which country are you studying ?

[drowsy quest]

us

[final pasture]

it makes me think of people liking pure maths that joins some engineering school after their prepa even though there's pretty much no pure math in the said school because they would feel bad if they did 2 year of prepa to finally join a math univ (since math univ is accessible directly after hs )

(I'm not saying it's your case)

[drowsy quest]

lol

my case is like, i liked probability, i like statistics enough, and i can get into a much better statistics department than math department

and job prospects are better for statistics tbh, both in industry and academic

[final pasture]

Fair enough

[viral jolt]

I was not that good at maths even tho I liked it, I reached my 'limits' in that sense during the preparaotry classes

[final pasture]

Imo prep classes are just bad at helping people developing their potential in math, anyone that needs time to develop intuition wouldn't be able to "be good" in prepa, even though they could be really good (and I mean even better than fast people) with enough time

I know someone like that

He did prepa and he was 2nd starting from the bottom during the first year in maths

and the prepa wasn't even good

so he left

and he joined univ

[viral jolt]

that is true, we don't have enough time, a chapter per week is overwhelming

[final pasture]

He's know doing his phd in category theory

[drowsy quest]

so are you trying to skip prepa and attempt the entrance exam?

[final pasture]

That's what I'm planning on doing yeah

[drowsy quest]

ooooo good luck

[viral jolt]

Focus on highschool, u still have history/french to deal with 😎

[final pasture]

last time I was in my high school

like physically

was maybe

5 months ago

and that's not because of covid

my last "bulletin" (idk what's the english word, h. please translate it for me lol ) is literally just zeroes and "wtf he's never here who's that ?"

and some of my prof

aren't even joking

Since I've literally never been present in their classes

[viral jolt]

'would say report'

[proud bear]

attendance?

[final pasture]

yeah, report with grades in it

[viral jolt]

C1 fluent 😎

[drowsy quest]

does that mean youre failing high school

[final pasture]

A1 fluent 😎

[drowsy quest]

does it matter what your grades are in highschool or only the entrance exams matter?

[viral jolt]

soon A1* then Shika

😎

[final pasture]

Yes

[final pasture]

well

[viral jolt]

to enter in uni, they don't really care

[final pasture]

at the end of hs

you get some diploma

to atleast be allowed to enter uni

this diploma

is easy

like really easy

and that's not even flexing

there's something like 95% of success

But you need it

and well

I am never attending at my classes

so I repeat classes

I'll probably repeat the same class for the 2nd time this year

(so next year I'll take the same class for the 3rd time )

But I had a year of advance, so that's only be the first year where I'm actually a year older that then people with me

[viral jolt]

Why don't you try to attend the exam this year, as a free candidate, so you can go to uni faster ?

[final pasture]

I can do it this year, but that's exactly what I'm planning on doing next year

[drowsy quest]

soo isnt it a mistake to be not going to school? lol

[viral jolt]

Not really, high school has really low expectations in france

in maths/physic

[drowsy quest]

university admissions are different in the US lol, if you fail classes in high school you're fucked

[final pasture]

it is a mistake, somehow, but when I try to force myself, it huh.. doesn't quite work

[viral jolt]

we are one of the worst countries to be honest

[final pasture]

yeah, hs in France really sucks

[viral jolt]

that's why we have Exams after two years of preparatory classes

an exam means equal chances

for anyone

and to be fair we go from being the zero in hs, to top maths uni later on

[final pasture]

that's why preparatory classes are so intense btw

[viral jolt]

so that's not the potential of the excellent students that is controversial but more the politics set up

https://www.cpge-paradise.com/MP4Math.php

what u do in one year

in maths only

[drowsy quest]

so you didn’t learn any math in hs you learned it all independently ?

[final pasture]

and there's also

english

physics

[viral jolt]

chem

[final pasture]

chemistry

french

[viral jolt]

sport for X 😂

[final pasture]

yeah lol

I mean

42 chapters

[final pasture]

Yes

I mean

[viral jolt]

• first year

in addition

[final pasture]

when I started math on my own

I wasn't even able to add two fractions properly

(not even joking )

I already had dropped out of school for 3 years, when I started learning maths, so my level was roughly the level of someone that stopped school in 8th grade + that forgot everything he learned before 8th grade (including 8th grade)

[viral jolt]

(going to sleep, have fun guys)

[final pasture]

good night

Anyway

[chrome hinge]

Hi there, can i get some help with a galois theory problem?

let be p a prime, define K=Fp(t), then f=x^p-x +t is irreducible in K[x]. I need to show that K[x]/(f) is a Galois extension of K

I would just like to know where to start... should i use the definition?

[compact needle]

Yes. What is required for an extension to be Galois?

[chrome hinge]

To be normal and separable

K has characteristic p

[compact needle]

That's true. Do we have a good guess about what polynomial should split in K[x]/f?

[chrome hinge]

Hmm

Im actually quite confused about this field... like, i dont have any intuition of it

[compact needle]

Well, in general, if f(x) is irreducible over any field K, the normal closure of K[x]/f is the splitting field of f(x). So if K[x]/f is normal, it should be the splitting field of f(x)

[chrome hinge]

Oh

So the guess would be f and we should try to prove it does split there indeed

[compact needle]

Yup. So we want to try to understand how the roots of f(x) are related

[chrome hinge]

But before going any further

is this extension finite?

[sturdy marsh]

yes

finite over K

[chrome hinge]

Over K, yes

hmmm

[sturdy marsh]

this is a pretty funny polynomial

[chrome hinge]

why? lol

[sturdy marsh]

you'll see when you figure it out

[sturdy marsh]

once you do this

[chrome hinge]

Ok

so suppose its finite

(i dont see it rn)

[sturdy marsh]

if K is a field

[chrome hinge]

I mean, i get its probably finite

[sturdy marsh]

and f is a polynomial

over K

then K[x]/f is always finite

[chrome hinge]

ooh

but how can it be an extension of K then

Since K is not finite...

[compact needle]

It is finite dimensional over K

[sturdy marsh]

oh I meant that it's a finite extension

[chrome hinge]

Oh ok

Nice

[sturdy marsh]

are you asking if the field is finite?

[chrome hinge]

Nope, the extension

[compact needle]

Just to be clear, it is finite in the sense that it is a finite dimensional vector space over K. But as K is not finite, the extension field is not a finite field.

But this is what people mean when they say that a field extension is finite

[chrome hinge]

Oh, yeah. I get that perfectly, thanks

Its just he said the field was finite, meaning the extension, and i got confused

[chrome hinge]

here

[compact needle]

He meant it is a finite extension

[chrome hinge]

Yep, now i know that

Is there something wrong?

I know what it means an extension to be finite

[compact needle]

I think we are all on the same page now

[chrome hinge]

Nice

So, ok, lets assume it is finite, i can probably show that with no problem

[compact needle]

So the question remains, if one root of f is in K[x]/f, why are all of the roots there?

[chrome hinge]

Hmm

Thats the adjunction ring (field in this case) of any root of f

Oh sorry, it is isomorphic to the adjunction of any root

It doesnt mean it has all roots

[compact needle]

Not in general. But it should when the extension is normal

[chrome hinge]

I know the class of x is a root in there

[compact needle]

That is true. We want to try to understand how f(x) factor.

Let's try a different problem (that I think is more intuitive). I claim that if f(x)= x^(p-1)-t instead, the claim is still true, that K[x]/f is Galois

[chrome hinge]

Is it irreducible too?

[compact needle]

It is

[chrome hinge]

Excelent

Ok, so we want to prove it is normal

And the guess for a polynomial that splits would be again f

[compact needle]

Yup. But here we can more obviously write down what the roots are

[chrome hinge]

yeah, they are all t^{1/p}

oh sorry

t^{1/(p-1)}

[compact needle]

What exactly do you mean by that? Is there only one?

[chrome hinge]

Hmm, i dont really know to be fair

since t is a variable...

[compact needle]

Let's just pretend p=3

[chrome hinge]

i thought there would be just one root, repeated p-1 times

[compact needle]

How many solutions are there to x^2-t?

[chrome hinge]

and therefore the extension would not be separable

two

[compact needle]

What are they?

[chrome hinge]

t^{1/2} and its additive inverse

[compact needle]

how about x^3-t?

[chrome hinge]

hmm

t^{1/3}

and idk the other two...

Like, we cant talk of i here can we?

[compact needle]

I claim that we shouldn't think of t^1/2 and -t^1/2 as additive inverses, but rather as t^1/2 and (-1)t^1/2

Here, what we really care about is that (-1)^2=1

[chrome hinge]

Isnt this just the same thing?

[compact needle]

Technically, yes. But for the cube example, what we care about are the solutions to x^3=1

[chrome hinge]

Oh, ok i get what you mean

[compact needle]

And you are right, that we don't necessary know that K has cubic roots of unity

[chrome hinge]

Yeah, thats my concern

[compact needle]

But what about the p-1 roots of unity in F_p?

[chrome hinge]

Ooooooh

[drowsy quest]

If a homomorphism is two-to-one, does that mean its kernel necessarily has two elements?

[chrome hinge]

Its just Fp

[compact needle]

Well, it's Fp^*

But exactly

[chrome hinge]

Ok, yes, youre totally right hehe

i forgot about the zero

[compact needle]

So what are the solutions to x^(p-1)-t?

[chrome hinge]

Ok yes its fermats little theorem

[compact needle]

for sure

[chrome hinge]

Humm they would be t^{1/p-1} times any element of Fp*

[compact needle]

So if I have one root of x^(p-1)-t in an extension of K, do I have the others?

[chrome hinge]

Yes, since Fp is within K

[compact needle]

Yup!

[chrome hinge]

Youre a genius :D

[compact needle]

So do you believe that K[x]/x^(p-1)-t is normal?

[chrome hinge]

I just need to know there is a root in there...

the class of x is a root

[compact needle]

Indeed

[chrome hinge]

I feel it kinda weird, but i suppose it makes sense

Like, the class of x is not in the form we were just talking about

But yeah, it is like an artificial root of f

and er is this the reason why it is separable too?

Like, the roots are all different

[compact needle]

Since you know that the units of Fp are distinct, you get p-1 different roots

So yeah, you have seperability

[chrome hinge]

Nice...

[compact needle]

You can also check the gcd of f and f'

[chrome hinge]

the derivative test

[compact needle]

Yup. But f'=(p-1)x^(p-2) whose only root is 0, so it's relatively prime to f (which does not have 0 as a root)

[chrome hinge]

Nice

and to extend to the x^p-x+t case... should i solve that equation and multiply by the elements of Fp* or something?

[compact needle]

The symmetry is different here. But the idea is similar

(hint: try addition instead of multiplication)

[chrome hinge]

Ok

its just the previuos equation was way easier to solve

[compact needle]

That's why I did it first 🙂

[chrome hinge]

Now i see

has it something to do with the frobenius automorphism?

[compact needle]

Here a question: if alpha is a root of f(x), what is f(alpha+1)?

[chrome hinge]

oh

its zero too

[compact needle]

🙂

[chrome hinge]

Because of char = 0

[compact needle]

yup

[chrome hinge]

:D

[compact needle]

And are alpha and alpha+1 equal?

[chrome hinge]

No, that would imply 1=0

[compact needle]

Indeed

I claim you have enough to show the Galois-ness of the extension

[chrome hinge]

Ok, i think thats pretty much it

Exactly hahaha

Your claim is certainly right

I appreciate all your help, thanks!!!

[compact needle]

You're welcome!

[compact needle]

Yes

[drowsy quest]

do you know a reason for this?

[chrome hinge]

Yes, that means theres only two elements that map to the identity

which is the definition of the kernel of an homomorphism

[drowsy quest]

oh, of course. i feel stupid hahaha

[chrome hinge]

Dont worry haha

[drowsy quest]

ty

[chrome hinge]

Np :D

[bronze talon]

Hey all, I've been tasked to do the following, given a cube centered at the origin of R^3. We consider the midpoints of each of the 12 edges of the cube. I'm tasked with verifying that these 12 vectors form a root system.

I'm sure this is trivial, but I can't even really think of what these vectors look like numerically

[chilly ocean]

Well, if we assume the vertices are (\pm 1, \pm 1, \pm 1), then midpoints would be simply computed.

[tight otter]

Let $\gamma$ be a root of the polynomial $x^2+1$. Consider an element $a\gamma+b\in\mathbb{F}_3(\gamma)$ with $a,b\in\mathbb{F}_3$ and $a\neq0$. What is its inverse in $\mathbb{F}_3(\gamma)$?

[cloud walrus]

panoramatopia

[tight otter]

i've calculated the inverses for each of the six elements described here

but i'm curious if there's a way that's more technical and calculates some generic form for all ay+b

[rustic crown]

(a gamma + b)*(a gamma - b) should do the trick?

[tight otter]

yeah thats what i got for all my results i wonder if thats sufficient

yeah cuz

a^2=1 in F3

regardless of what a is

[rustic crown]

that would be (a^2 gamma^2 - b^2) = -(a^2+b^2) = 2-b^2 which is not 0 as b^2 = 0 or 1

[tight otter]

so you'd get g^2-b^2=1 or g^2+1=0 which follows the polynomial

[rustic crown]

(a g - b)/(2-b^2) is the required inverse

[tight otter]

mmm yeah you're right good catch

just noticed that looking at my individual inverses bc gamma*2gamma=0

[rustic crown]

=1?

[tight otter]

braindead

sorry

yes

i stg my brain drops packets when i'm trying to talk about math

thank you for the help

[rustic crown]

yw

[tight otter]

just a quick logic check--its sufficient to say that the order of an element $a$ in a cyclic group is $p_1^np_2\cdots$ as long as $a^{p_1},a^{p_1^2}...\neq1$, correct? e.g.--if $a^9=1$ and $a^3\neq1$ then the order of $a$ is 9

[cloud walrus]

panoramatopia

[rustic crown]

are you checking over all divisors?

consider the group of units mod 7 that is, {1, 2, 3, 4, 5, 6} under multiplication modulo 7
can i say order of 2 is 6 = 2*3
because 2^2 is not 1 but 2^(2 * 3) = 1?

[tight otter]

yes, checking over all divisors. in your example it wouldn't be true that the order is 6 because 2^3=8=1 mod 7

[rustic crown]

yep, then its fine

you can reduce your work a bit

a has order n if a^(n/p) is not 1 for any prime divisor of n and a^n = 1

[tight otter]

great!

the example i was trying to show was just that ord(a) = 8 because a^2, a^4 != 1 and a^8 = 1

just wanted to brush up on my logic there

[tight otter]

Sorry for another question but a classmate and i are stuck on this one problem: Show that at least one element of the set ${2,3,6}$ is square in $\mathbb{F}_p$ with $p\geq7$. So far we've supposed by contradiction that none of them are squares. Because $\mathbb{F}_p^\times$ is cyclic, then it has a generator $a$. Because none are squares, then for all $1\leq x,y,z\leq p-2$, it's true that $a^{2x}\not\equiv 2$, $a^{2y}\not\equiv 3$, and $a^{2z}\not\equiv 6$, all mod $p$. By extension $a^{2(x+y+z)}\not\equiv 36$ mod $p$. If we pick $x+y+z=p$ (which we can do because as shown $x$, $y$, and $z$ are all arbitrary) then we obtain $a^{2p}\not\equiv 36$ mod $p$ or $a^p\not\equiv 6$ mod $p$. By Fermat's Little Theorem, this states that $a\not\equiv 6$. This is a pretty easy counterexample in some fields but I'm not sure how to extend this upwards to all $\mathbb{F}_p$, and I'm also not sure if we made a leap in logic somewhere with all the incongruences

[cloud walrus]

panoramatopia

[chilly ocean]

that sounds like a weird problem

[tight otter]

yeah fuck me for having a number theorist as my algebra professor

[rustic crown]

It's actually a pretty cute problem

Generally try NOT to manipulate incongruences, as you might make a "leap in logic"

[tight otter]

yeah that's what I was worried about

[rustic crown]

like 4 is not 1 mod 7

2 is not 1 mod 7

but 2*4 = 8 = 1 mod 7

[tight otter]

mmmmm yeah that's what i was worried about

[rustic crown]

i can give you a proof with both using and not using the fact that multiplicative group of Fp is cyclic

which one do you want?

[tight otter]

I'm a bit more interested in the one that uses its cyclic properties, as that was the hint my professor gave

[rustic crown]

okie so try to think about this. Say a was a generator.

Show that a (non-zero) element b is perfect square if and only if b = a^(2x) mod p

a^2x is clearly a perfect square, so what you need to show is, the other direction. That is if the exponent is not even, then the element isn't a perfect square. That is, a^(2x+1) can't be b^2 for some b.

[tight otter]

Well, clearly b=a^r mod p for some r

[rustic crown]

yep

[tight otter]

and because b is a perfect square, then you can take the square root of both sides and you should get a number that exists in F_p, that is, a^(r/2)=a^q mod p

Square both sides and you get a^r=a^2q mod p

[rustic crown]

but are you saying r is even in this case?

how do you just take square roots?

if r was odd then a^(r/2) doesn't make sense. Also there are 2 square roots for a number, which one do you use?

[tight otter]

good point

hmmmmm

[rustic crown]

more precisely i want you to show that something is not square if and only if the exponent would be odd

[tight otter]

alright

clearly if it's not square then the exponent is odd

because an even exponent implies a square, as you've shown

[rustic crown]

yea so say a^(2x+1) = c^2 then c = a^y and hence we'll get a^(2x+1) = a^(2y) is this bad?

[tight otter]

if we have a^r=b mod p with odd r, then we can multiply both sides by a to get a^(r+1)=ab mod p. Clearly r+1 is even so ab is a perfect square then. However, this would imply both a and b have a square root, but a has odd order because it generates a group with prime degree.

uhhhh but then that circles back to the beginning of the sentence lol

and yes that is bad

because then you're saying that an even number is equivalent to an odd number

mmmmmmm

[rustic crown]

how does ab being square imply both a and b have a square root?

[rustic crown]

there is a missing step which lets you conclude this

[tight otter]

it doesn't, disregard my logic

well, if both 2x+1 and 2y are less than p then the problem is done, because all powers of a under a^p are distinct

[rustic crown]

yea so the problem is this would tell us that
a^(2(x-y)+1) = 1

yep that's one way to say it

but generally you'll see this argument more. order of a is (p-1) hence (p-1) divides 2(x-y)+1

[tight otter]

mmmm

but p-1 is even

[rustic crown]

but if p is an odd prime then i get 2 divides 2(x-y)+1

[tight otter]

and 2(x-y)+1 is odd

i see

alright, so we've shown that a perfect square must be an even power of the generator

[rustic crown]

yea so what we proved is a number is perfect square if its some even power of the generator and its not a perfect square if its odd power of the generator

[tight otter]

i guess the logical next step would be to prove that 2, 3, and 6 can't all be odd powers of the generator

ohhhhh

[rustic crown]

if either of 2 or 3 are perfect square then we're done!
and if they are not, 2 = a^r and 3 = a^s with r and s odd

[tight otter]

i think i'm seeing it here

yeaaaaaaaah

damn that's a swifty solution!

[rustic crown]

this gives 6 = a^(r+s) but (r+s) is even hence in this case 6 is a perfect square

What is generally done is, with this you can define

$\left(\frac{a}{p}\right) = \begin{cases}0 & a \equiv 0 \pmod p\ 1 & a \text{ is a quadratic residue}\-1 & a \text{ is a quadratic non-residue}\end{cases}$

[cloud walrus]

det

[rustic crown]

quadratic residue is just a fancy name for perfect square

with what we just prove, you can write it neatly as

$\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$

[cloud walrus]

det

[rustic crown]

(just to be clear, these are not fraction, its just a notation. its called the Legendre's symbol)

if a and b are not perfect squares then their product is.

[tight otter]

oooh that's fancy

so you could extend this same logic to any set {p, q, pq}, right?

[rustic crown]

yep

[tight otter]

wow

[rustic crown]

one nice thing you get from the problem is this

[tight otter]

i was wondering what the significance of 2, 3, and 6 was

[rustic crown]

consider the polynomial

$(x^2-2)(x^2-3)(x^2-6)$

[cloud walrus]

det

[tight otter]

oh yes! that's the second part of the problem

i did that one already

[rustic crown]

oh lol

[tight otter]

also very cool

it has roots in all F_p but no roots in Z

[rustic crown]

yep!

another fun one was this

[tight otter]

• at least one root that is

[rustic crown]

16 is a perfect 8th power modulo every prime, but its not a perfect 8th power in Z

you can give it a shot if you're bored

[tight otter]

hahaha i very well might

[rustic crown]

its same as saying (x^8 - 16) has a root in any Fp but not in Z

[tight otter]

yeah that's really neat

thank you for all the help and explanation det

[rustic crown]

[stark obsidian]

hi, could someone recommend me a book or document to read about properties of homomorphisms between complex tori?

[golden pasture]

alternatively, isogenies between elliptic curves?

[stark obsidian]

Yes, something like that, but not too advanced

[sharp sonnet]

how did you define ideal?

[lunar coyote]

ideal is the integers

[sharp sonnet]

what?

[lunar coyote]

i'm confused

[nova plank]

Show the whole question please

And give your definition of an ideal of a ring

Okay, definition.

[lunar coyote]

an ideal is a subring which satisfies if a,b is in R then a-b is in R and say r is in the ideal then ra=ar is in R

[lunar coyote]

is that the definition you wanted?

[nova plank]

Yes

Now if you want to show a subset, take an element of the smaller set and show it belongs in the larger set

It follows like immediately from the defn

Can you write it out properly

[lunar coyote]

so would the smallest integer be just n

because when you multiply n.z the smallest positive integer would be 1*n = n

[nova plank]

n can be any element of I

But it is the smallest one in this case

Since they say that in b

[hidden haven]

Ideals are not subrings

[old lava]

ideals very much are subrings

they just so happen to satisfy conditions even stronger than ones for being a subring

they're special subrings

[compact needle]

That depends on your definition of "ring"

[hidden haven]

If you take unital rings then no

[old lava]

well I don't do the ring/rng fuckery

[hidden haven]

They are only subrings if you are working with rngs and that's certainly not standard

[compact needle]

Nonunital rings seems more common in the theory of noncommutative rings. But in commutative settings, I almost always seen ring to mean unital, as well.

[hidden haven]

Even in non commutative rings I've only seen unital being assumed

[compact needle]

I've seen some noncommutative geometry people explicitly want nonunital rings. For what it's worth, Dummit+Foote do not require ring in their basic definition, so this definition definitely does persist at large

[hidden haven]

I see

[inner acorn]

Hmmm, I'm trying to find two finite abelian groups G & H such that:
C[G] ≅ C[H] as complex group-algebras
but R[G] ≇ R[H] as real group-algebras

And ngl.. I'm kinda stuck n.n;;

I think if I pick G & H to be finite with the same order, then C[G] ≅ C[H] would follow immediately

But I don't really know how to mess with R[G] ≇ R[H]

[nova plank]

If you know there is a convention where ideals are subrings and someone says that is their definition, idk why you would reply to them that ideals aren't subrings

[hidden haven]

I wasn't even aware there was a convention where rings are actually rngs

Because I have only ever seen unital rings

[compact needle]

Consider the answer to this question, and see if you find it inspiring: https://mathoverflow.net/questions/229213/a-group-algebra-isomorphism-problem

[inner acorn]

Hmmm, well all the criteria is automatically upheld except for nth roots of unity.
Which to me, suggests I should pick C4 (so roots would be 1, -1, i, -i) and C2 x C2 which does have all it's roots in R.

So I'm guessing R[C4] ≅ R x R x C
and I'm guessing R[C2 x C2] ≅ R x R x R x R

Is this what you were hinting at by posting this link?

[compact needle]

yup

I'm not actually sure about the isomorphism type of R[C4]. But it does have an element of order 4, which R[C2xC2] does not

[inner acorn]

ty, I'll argue the orders of elements, it seems to assume less

[rigid cave]

I am reading a proof about the simplicity of A_n. Is the red part in the image really necessary? Would the proof still be valid if we just ignored it?

[chilly ocean]

In a ring, we have the distributive property a * (b + c) = a * b + a * c, where * is the multiplication in the ring and + is the addition in the ring. But * does not necessarily have to be commutative

However, we must also have (b + c) * a = b * a + c * a be satisfied.

So I am confused.

Or is it that they have to be satisfied, but it is not necessarily the case that a * (b + c) = (b + c) * a?

[carmine fossil]

You don't get a*(b+c)=(b+c)*a

[chilly ocean]

So it's the latter of what I wrote?

[carmine fossil]

Yes

[chilly ocean]

They're satisfied, but not necessarily equal

Okay thanks

[rigid cave]

I mean, I can surely find a tau that doesn't commute with sigma and that fixes both sigma(i) and i without having to know that all the cycles in sigma have the same length, right?

[rustic crown]

yea looks good to me

lemme think again tho

it could just serve as a tool to reduce the number of cases that you need to consider to show that such a tau indeed exists?

[lunar coyote]

for this question can i just prove it by just stating the def of the ideals

[rigid cave]

So we want a $\tau$ such that $\tau \sigma \tau^{-1} \neq \sigma$. Let $\sigma = (a_1 a_2 \cdots)(b_1b_2 \cdots)$. Then $\tau \sigma \tau^{-1} = (\tau(a_1)\tau(a_2) \cdots)(\tau(b_1)\tau(b_2) \cdots)$. The lenght of $\sigma$ is greater or equal to 6. Tau must fix 2 elements, which means that 4, at minimum, must be permuted. But tau is a double transposition so we could always find a tau that doesn't commute with sigma. I really don't understand why the cycles need to have the same length...

[cloud walrus]

older sister

[rustic crown]

(just to confirm, by double transpositions we just mean product of 2 transpositions and not product of 2 disjoint transpositions, right)

[rigid cave]

Yes, I mean tau = (ab)(cd) where a, b, c and d are different

[rustic crown]

but then what if n = 5, sigma = (12)(34)(56)?

none of (34)(56), (35)(46) or (36)(45) would work.

[rigid cave]

But how do you know that the above sigma is in the normal group N?

[rustic crown]

i can't know that 😶

we're trying to show that N is trivial

[rigid cave]

yeah but your sigma is not even, right?

[rustic crown]

ooh oopsie

lol

[rigid cave]

It seems so weird because I can't find any reason why knowing that all the cycles in the decomposition of sigma has the same length would help. It also feels weird for a proof to have unnecessary information...

[lunar coyote]

if i have some u,v

but u and v aren't multiples of each other

would the gcd(u,v) be the smallest postive integer

[rigid cave]

It would be 1, I think, because u and v are relatively prime

[lunar coyote]

could i say that 1 is say d or something arbitary

that belongs to an ideal in the integers?

[rustic crown]

i'll probably just assume its there to help you find such a tau? for instance I could take 2 cases sigma(sigma(i)) = i and sigma(sigma(i)) = j which is not i. In the second case we can consider tau to be (sigma(i), a)(b c) where a and j are different and in the first case that helps a bit to see how sigma would look like, it will just be a product of 2 cycles.

[rigid cave]

yeah that might be the case, I am not entirely sure

[rustic crown]

what about 4 and 6

[rigid cave]

oopsie

[lunar coyote]

triyng to something likt

a is an element of an ideal where a is the smallest positive integer

[rigid cave]

Then a generates that ideal I think if a is the smallest positive integer in the ideal, assuming we are talking about ideals in Z