serious-discussion

[jaunty ibex]

shape is literally a physical thing lol it has volume etc

[dreamy ore]

ykw

[solid snow]

????

[dreamy ore]

i simply cannot be assed

[jaunty ibex]

I won't talk about it either

[umbral thorn]

what question?

[dreamy ore]

maybe try reading the chat!!!

[jaunty ibex]

But anyhow my point was you are working to solve a world problem

The end result is something describing the physical quantities (size) of something

[dreamy ore]

ok here is the equivalent question

orthographically project an ellipsoid onto a plane

[solid snow]

is trig physics now

[umbral thorn]

kindly send question

[dreamy ore]

no

im not asking for help

[jaunty ibex]

This is very elitist btw deltoid

[umbral thorn]

juat show me question

[latent edge]

???????

[dreamy ore]

there is no question

its a simple curiosity

"determine the figure of a planet"

[latent edge]

It's literally the same thing with how some people don't consider Spec Z as a geometric object

[umbral thorn]

okhy

[dreamy ore]

thats it

[jaunty ibex]

Spec Z is not geometric though

[umbral thorn]

The actual shape is called an oblate spheroid.

[junior veldt]

Oh no AG

[dreamy ore]

??????yes i know??

its a kind of ellipsoid

ok stayX what is ur point

what r u trying to do

[umbral thorn]

i just trying to solve ur question

[jaunty ibex]

Why sully

[dreamy ore]

i will solve it on my own thanks

[jaunty ibex]

Spec Z is not geometric lol

[elfin smelt]

having a topology is pretty geometric

[umbral thorn]

as u wish

[latent edge]

maybe read hartshorne ch2

[junior veldt]

disagree

[umbral thorn]

i just tryied

[dreamy ore]

idk how deltoid is making the point that a problem involving ellipsoids and planes and projections is not geometric

but ykw

[latent edge]

it is geometric lol

[dreamy ore]

you do you

[junior veldt]

to me geometric means something like riemannian/lorentzian geometry

[latent edge]

i just meant in the pure sense it's not really that

[junior veldt]

where you care about lengths and angles and stuff

[elfin smelt]

like it has to have distance?

ah

[junior veldt]

topology is something separate in my mind

[elfin smelt]

that makes sense

[junior veldt]

I am coming at this from a physics background ofc

the geometry of spacetime is a lot more information than the topology of spacetime in GR

[latent edge]

took me a while to ditch the idea that the only "real" geometry is euclidean

[dreamy ore]

the problem in full is just this

given an ellipsoid

u are looking at it from some angle

[latent edge]

yes i agree this is geometric af

[dreamy ore]

and the sun is shining on it from another angle

[raven plaza]

I see three ellipses on a plane

[dreamy ore]

determine the shape of the shaded area

[latent edge]

but once you put real life context in it, i dont call it pure math at all

[junior veldt]

I haven’t really done much AG but it’s very hard to see how it is at all geometric

[dreamy ore]

i never called it pure math

i called it geometry

[latent edge]

its okay im just used to pure stuff

[dreamy ore]

it is a geometry problem

u are just obtuse

[jaunty ibex]

okay shiru ur being an asshole now

[dreamy ore]

wouldnt be an asshole if he didnt jump to hate on what im doing

[solid snow]

the variable names make it not pure math

[dreamy ore]

calling it a waste of time

[latent edge]

i know a lot of people meme about the spec Z diagram but it is geometric

[solid snow]

[dreamy ore]

also this

id still consider this pure math

[latent edge]

[junior veldt]

Again I just don’t see how

Tbh I don’t even know what spec Z is

[latent edge]

the points are the prime ideals of Z

[junior veldt]

@.@

[elfin smelt]

it’s topology isnt hausdorff which makes a lot of people think its even less geometric

[junior veldt]

im already lost

[raven plaza]

AG is still mostly about polynomials over a field

[latent edge]

yeah i know non-hausdorff stuff make people go away

[elfin smelt]

you should probably see the picture if you haven’t, but know it definitely isnt very enlightening

[latent edge]

LMFAO

[dreamy ore]

god i hate how elitist this server is

[junior veldt]

Yeah I guess I just don’t care about polynomials that much

[dreamy ore]

why do yall hate on all math that isnt glorified set theory

[raven plaza]

Fair enough

[junior veldt]

I don’t!

[dreamy ore]

theres still math to be done

[latent edge]

that's fair

[solid snow]

in all fairness i dont think this is a representative view of a lot of the server ime

[latent edge]

im gonna find a reason for u to care about polynomials

[junior veldt]

I guess, what I like about geometry is that it’s very concrete and visualisable

[dreamy ore]

"trig isnt math its physics" is the most absurd take ive ever heard

[junior veldt]

And AG to me feels like “ok but what if you threw away the pictures and just focused on symbol-pushing “

and im like

ok you can but why should I

[raven plaza]

"Geometry" refers to "space with a sheaf of functions on it" I think

[junior veldt]

@.@

[junior veldt]

Yeah this is weird

[latent edge]

sure but like you can still get pictures

it just gets cursed I agree

[junior veldt]

Right but for me geometry is pictures

[dreamy ore]

also deltoids take basically saying euclidean geometry isnt high level enough for me to even consider it as geometry

this server is very weird

[junior veldt]

I mean it has geometry in the name

I’d say it’s geometry

[raven plaza]

Euclidean geometry is not real math

[latent edge]

???

[raven plaza]

Euclid did not know about ZFC+AxG+Collatz+RH

[junior veldt]

It absolutely is what

Ah I see you were memeing :3

[latent edge]

by that claim this should be geometric

[junior veldt]

You misunderstood the implication direction

Geometry is pictures

This doesn’t mean pictures is geometry

[latent edge]

hmmmm

[raven plaza]

I have a geometry of your mum

[latent edge]

higher dimensional stuff don't have clear pictures

[junior veldt]

Sure they can

[latent edge]

even in like euclidean geo

[junior veldt]

I don’t think you’re gonna be able to convince me this is geometry lol

I’m not an algebra supremacist :)

[latent edge]

im not algebra supremacist either

[junior veldt]

sure..

[umbral thorn]

i can

tell me

[junior veldt]

you can try I guess?

[umbral thorn]

yes sure

[jaunty ibex]

Cant tell if ur memeing or drunk

[umbral thorn]

@junior veldt what is the question about

[jaunty ibex]

Idk geometry for me is something closer to differential geometry

[raven plaza]

^ @umbral thorn

[junior veldt]

yeah same

[raven plaza]

They literally replied to this lol

[jaunty ibex]

This is wrong. Deltoid didn't say Euclidean geometry is not geometry

[twin flicker]

everything is set theory trust

[junior veldt]

Technically true

[dreamy ore]

👍

[raven plaza]

This picture depicts the scheme morphism Spec Z[φ] → Spec Z where φ is the golden ratio

[junior veldt]

huhhhhh

[umbral thorn]

hmmm hmm tough ?

[raven plaza]

Wrong, this is geometric because the golden ratio appears geometrically in nature

Also !nogpt

[umbral thorn]

might be

[raven plaza]

!nogpt

[quasi jetty]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[jaunty ibex]

[junior veldt]

jojo part 7 be like

[umbral thorn]

this questions come from ?

[jaunty ibex]

Does this actually help though?

My approach to number theory has been just algebraic so far

[raven plaza]

Idk ask deltoid

[latent edge]

it's depicting what primes in Z[phi] ramify, split or stay inert

[umbral thorn]

oh damn it was not satisfyin so far

[latent edge]

what is not satisfying

[raven plaza]

Ur mum

[umbral thorn]

i cant understand this just leaving 😆

[latent edge]

its okay i can explain with easier examples

[umbral thorn]

yes sure

[latent edge]

So you know how in Z (integers) we have primes 2,3,5,7,11,...?

[umbral thorn]

yes

[latent edge]

right

[umbral thorn]

yup

[latent edge]

Now what if we wanted to extend the notion of integers

And do number theory with different type of "Z" if you will

[umbral thorn]

are u lecturer ???

[latent edge]

just stay on point

[umbral thorn]

ok

[latent edge]

Anyhow we can take Z[i] which involves taking a+ib, for a,b in the integers and "i" is the imaginary number you might know

[umbral thorn]

yup i know

[latent edge]

okay cool

Now the next question to ask is whether you have unique factorization in this ring

and it turns out the answer is yes in this case

so you have the same analog of factorizing an integer into product of primes powers

times some unit

so far so good?

[umbral thorn]

oh okhay

tysm

[latent edge]

let me know if something doesnt make sense

okay

[umbral thorn]

TYSM

lets b friends

[latent edge]

now it's easy to see Z sits inside Z[i]

you can just let b=0 in a+ib and you get the integers

but then if Z[i] has unique factorization, are the primes in Z still primes in Z[i]?

[umbral thorn]

what about splitting

[latent edge]

we are getting into that in a min

what do u think

do you think all primes in Z stay prime in Z[i]?

[umbral thorn]

no

[latent edge]

yep you'd be right

in fact some primes stay the same, we call these "inert"

Some split like p=(a+ib)(a-ib). These are called split primes

and there is a third category which is incredibly weird

for example in Z[i], all odd primes are either split or inert but 2 is very special

[umbral thorn]

ya

[latent edge]

up to units 2 = (1+i)^2

(1+i)^2 = 2i so it isn't really 2

but "i" becomes a unit in Z[i]

just how {1,-1} are units in Z

so we treat them the same

[junior veldt]

Well isn’t 2 = (1 + i)(1 - i)

[latent edge]

yeah right

but you can rewrite it as (1+i)^2

[umbral thorn]

yes we can

[latent edge]

yeah so the point is since you can rewrite it like this, it's not really a split prime

in some sense it has a geometric picture of having "condensed points"

[latent edge]

which is similar here. 5 is the special case here and you can see that the dots are kinda dense

these are called ramified primes

[junior veldt]

Now I’m confused

[latent edge]

[umbral thorn]

5=(2+i)(2-i)

[latent edge]

yeah

[latent edge]

what's up

[jaunty ibex]

NT discussy yay

Thank god no modular forms

[junior veldt]

I don’t understand what you’re saying

[latent edge]

i was being kinda low level with them without getting into ring theory too much

[raven plaza]

Ramified means repeated prime

[junior veldt]

huh…

[latent edge]

do u want a ring theoretic version of what i said

[raven plaza]

2 = -i(1+i)²

So (1+i) is repeated here

[latent edge]

i think their confusion here is 2 = (1-i)(1+i) so it seems like 2 is split

[raven plaza]

No different than -4 = (2)(-2)

[latent edge]

yeah

[vernal iron]

makes sense

[jaunty ibex]

I gotta say tho number theory is a bitch to understand sometimes

[junior veldt]

I’m not sure this will even be helpful

You can try I guess

But I have 0% intuition for algebra

[junior veldt]

Yes but I wouldn’t call -4 a square

[long burrow]

can someone help me understand the math behind this answer?
https://math.stackexchange.com/a/2174924

Im struggling to follow this, if the area of a sphere is 4 * pi * r^2 then how does he get pi / 5 for the area of one triangle?

I only see up to:
4 * pi * r^2 = (pi * r^2) / 5

[raven plaza]

If anything is repeated it's ramified

[junior veldt]

At least in the reals

[junior veldt]

boop boop

now boop is ramified?

[long burrow]

where does the r^2 go, or since it's a unit circle, r is 1 and it cancels?

[raven plaza]

Like if x = y²zw

The ² makes it ramified

[junior veldt]

This is a googit moment for me

[latent edge]

what are you confused about? is it (1+i)(1-i)?

[junior veldt]

A googit is a collection of exactly 3 humans, 2 planets, 1 proton and 1 beaver

[latent edge]

[junior veldt]

I use example to illustrate the difference between “parsing” a definition and “understanding” a definition

[junior veldt]

Given this I can recognise whether a collection of things is a googit or not

So I can “parse” the definition

[latent edge]

okay let's start again. Are we okay with the fact that {1,-1,i,-i} are units in Z[i]?

[junior veldt]

But I do not “understand” the definition at all

[junior veldt]

In terms of why we would give a name to this concept

[raven plaza]

As in why we need to distinguish splitting and ramification?

[junior veldt]

Something like that

To me your definition of ramified reads like a googit

[latent edge]

if we are okay with the units then I claim that (1+i) and (1-i) are basically the same element up to units

[junior veldt]

Yes

[junior veldt]

I don’t understand this

[raven plaza]

They define the same principal ideal

[latent edge]

well how do you make 1+i -> 1-i?

yea

[thorn jay]

@shell vigil

Can I DM you

I don't know where to ask

[latent edge]

i(1-i) = 1+i and 1+i is irreducible

[shell vigil]

Kk

[latent edge]

so we can take one of them say 1+i

as principal ideal

[junior veldt]

Conjugation I guess

@.@

Why are we talking about principal ideals now

[latent edge]

because 1+i is principal

[raven plaza]

We're really considering factorizations of ideals

[latent edge]

yeah

[junior veldt]

See this is what I mean by having 0% intuition for algebra

I am utterly lost

[raven plaza]

(2) = (1+i)(1-i) = (1+i)²

[latent edge]

ok here is the intuition for ideals in a nutshell

[junior veldt]

This is not an equality

[raven plaza]

The bracket denotes principal ideal

[junior veldt]

Oh wow well that was suuuuuuper clear from the notation

[latent edge]

ideals are the "right" algebraic object to think about in a way that you dont get lost with unique factorization

unique factorization is well defined for these guys and it doesn't fail

[junior veldt]

I don’t think I am capable of understanding this

[umbral thorn]

thankuuuu sooooo much for this

[solid yarrow]

Same!

[latent edge]

when the ring is UFD the "ideals" can be thought of as just the elements

[junior veldt]

.

[umbral thorn]

u r so coool bro @latent edge

[junior veldt]

I really have no clue what you’re going on about

[latent edge]

are you familiar with ideals?

[junior veldt]

At this point I’m tempted to say no

[latent edge]

: (

[junior veldt]

I’m not sure if the words you’re saying mean what I think they mean

So I’ve essentially lost any ability to understand you

[umbral thorn]

@latent edge will you friendship withme plz

[shell vigil]

[umbral thorn]

if i have query i will ask u

[haughty lion]

Hey Obi

[vernal iron]

Average convo with Deltoid ngl (I almost used his actual name 💀 (5:30 am moment, gn chat))

[junior veldt]

I see…

This often happens when people try to explain pure math to me

It feels like they’re speaking a different language but using the same words I know

[vernal iron]

don't worry, it is hard to get and it is okay if you don't understand it

[junior veldt]

Mhm, I know

Though it gets annoying when people keep glazing how “intuitive” or “trivial” it is

[raven plaza]

The perfect life is attained through marriage

Because you need rings to have ideals

[vernal iron]

💀

[latent edge]

im basically trying to say that if you think of regular elements as primes in different ring of integers, this is very dangerous because unique factorization doesn't hold in general

[shell vigil]

Best explanation fr

[junior veldt]

I don’t understand any of these words

[umbral thorn]

idle

[latent edge]

i will give an example

say we are in Z[sqrt(-5)]

[junior veldt]

none of these words either

[vernal iron]

so Z is a ring right, if you look at those primes in a different ring, they might not have unique factorization

[junior veldt]

I don’t know what that is

[latent edge]

bruh

it's the set of elements a+b sqrt(-5) where a,b are integers

[junior veldt]

yeah not every ring is a UFD

[junior veldt]

I don’t know what this is

[latent edge]

it's a ring

[vernal iron]

similar to C being a + bi, where a and b are real numbers

[junior veldt]

I don’t know what that is

[latent edge]

use noggin

[junior veldt]

don’t know what you mean by that

[vernal iron]

except it is done with Z, and instead of i it is 5i

[vernal iron]

he means think about it

[jaunty ibex]

I think deltoid is losing his shit now

[junior veldt]

I’ll have to take your word for it

[vernal iron]

lol I already asked him what he meant

[shell vigil]

Is that pfp a nova

[junior veldt]

it’s a nebula

[shell vigil]

Ic

[latent edge]

anyhow here is why you should care about ideals. In Z[sqrt(-5)] we have that (2)(3)= (1-sqrt(5))(1+sqrt(5)).

[junior veldt]

I assume you’re not talking to me here

[latent edge]

that doesn't go with the idea of unique factorization

I am

assuming

[junior veldt]

ok I still don’t understand anything you are saying

[latent edge]

talking to u

i know i may be drunk rn but cmon rack noggin

[junior veldt]

I am very confident that I am incapable of understanding this right now

[junior veldt]

also don’t know what this means

[raven plaza]

Drunken man injects man-made horrors beyond comprehension in the likes of Hartshorne into poor innocent souls

[jaunty ibex]

Depicts Deltoid

[junior veldt]

luckily I have no soul

[jaunty ibex]

Thank god I'm not infected by geometry

No diff geo no alg geo

Only algebra

[long burrow]

@latent edge could you chime in on #help-8 ?

[junior veldt]

I think my takeaway from this is I should stay the hell away from AG

[vernal iron]

True

[jaunty ibex]

Kinda makes me wonder

Deltoid used to stay away from AG

Now AG is in his blood

[junior veldt]

I am very sure that this will not happen to me

I have 0% intuition for algebra after all

[vernal iron]

Hartshorne happened

[jaunty ibex]

Hartshorne is bad

[vernal iron]

Chat has recommended it to me 💀

[jaunty ibex]

Idk I stay away from arithmetic geometry

I mean look

You have NT which is incredibly cursed

And algebraic geometry

what a cursed combination

[umbral thorn]

my father is math teacher'

[jaunty ibex]

I know deltoid gets sad when he explains but the one trying to understand doesn't get it

[shell vigil]

He gave up

[jaunty ibex]

But idk chat he already does dark magic at this point

[junior veldt]

Rip

[vernal iron]

He should go to bed ngl, it is 5:47 for him

[umbral thorn]

@latent edge is one of the best in chat

[junior veldt]

To be fair my intelligence drops to 0 when it comes to algebra

[shell vigil]

Maybe he did

[jaunty ibex]

I doubt

[vernal iron]

I hope

But I agree with Vero

I also doubt

[shell vigil]

Bros drawing a visual guide for her to understand the concept lmao

[vernal iron]

Potentially

[umbral thorn]

@jaunty ibex hi

[jaunty ibex]

I'm taken. Thanks

[vernal iron]

But there's good images online I think

[shell vigil]

Bro just joined today and randomly pinging people in chat

[vernal iron]

At least it is people who are talking currently

[shell vigil]

Watch them ping mniip now

[vernal iron]

Don't.

[latent edge]

ok yall here me out

This is geometric.

[vernal iron]

Hearing you out

[shell vigil]

He did!

[latent edge]

it's a lattice, I can visualize it

good enough

[junior veldt]

…no

[latent edge]

yeah it's not immediately geometric

[vernal iron]

Why are you sending a diagram with Galois theory on it for this chat

[junior veldt]

Don’t understand what this means

[latent edge]

im going with the symmetries -> can think of symmetries of some geometric object

[junior veldt]

idk what that means

[latent edge]

well take a square

[junior veldt]

idk what that is

[neat lintel]

faded

[shell vigil]

Destroyed

[neat lintel]

blooded

which evolution og evee is most powerful

[latent edge]

union of four line segments in R^2 if you will

[junior veldt]

not sure what this is

[latent edge]

ok bye

[junior veldt]

bye!

I do find it strange how persistent you were

[shell vigil]

Denied him too much

[junior veldt]

I mean you’ve gotta understand

[junior veldt]

.

So I literally could not trust any words out of his mouth

[jaunty ibex]

Deltoid bad tutor smh

L

[shell vigil]

It's okay i dont understand anything about what you guys are saying anyway i just finished high school lol, just here to gawk and marvel

[elfin smelt]

this lattice shows which permutation groups are in other permutation groups, and it does have some nice images

[junior veldt]

Nah I’m just stupid when it comes to algebra

Or pure math more generally

[elfin smelt]

also the first lattice in that image isnt of roots of unity, so it maybe isnt as pretty

[junior veldt]

I like this!

[neat lintel]

can i descuss ben 10

[shell vigil]

I have a ben 10 watch

But its broken

[elfin smelt]

(this isnt the same lattice, but it should have an associated one)

[neat lintel]

i want to know why ultimate echo beat kevin but coudnt beat agregor

[shell vigil]

Idk any lore lmfao i barely watched some episodes

[jaunty ibex]

I'm an algebraist. I would've done a worse job at explaining ngl

[shell vigil]

But theres surely benten nerds in disc 1

[junior veldt]

it’s difficult to see how it could’ve gone worse than that

[neat lintel]

ultimatric

[jaunty ibex]

taking elements in O_K is not great because it doesn't guarantee UFD so we instead think of ideals instead

[junior veldt]

@.@

I already dunno what O_K is

[jaunty ibex]

Since if O_K has class number 1 then it is okay but otherwise no lol

[jaunty ibex]

Examples: Z[i],Z[phi],Z

Basically what deltoid has been yapping about

[raven plaza]

Class numher

[junior veldt]

so rings…?

[jaunty ibex]

yeah they are rings

They come from a field extension Q(alpha)

[surreal sapphire]

they are the integers over some field

[jaunty ibex]

They are called ring of integers

[surreal sapphire]

the ring of algebraic integers

[jaunty ibex]

You can think of them as the intersection of Z and Q(alpha)

[elfin smelt]

isnt Z always contained in them though?

[jaunty ibex]

yes

[surreal sapphire]

yes, but uhh

its larger than Z

[jaunty ibex]

Z inside Z intersect Q(alpha)

[junior veldt]

Is alpha here some (algebraic?) complex number?

[junior veldt]

Oh I have heard of these!

[jaunty ibex]

Yes

[junior veldt]

Like isn’t Z[i] for the quadratic integers?

[surreal sapphire]

its always a finite Z module

ye

[junior veldt]

Ah so it’s like

You adjoin an algebraic integer to Z?

[jaunty ibex]

yes

[junior veldt]

Ok cool

[surreal sapphire]

you maybe know the notion of "being integral over a subring"

[junior veldt]

Hmmm so you can put a ring structure on the ideals?

[junior veldt]

No unfortunately

[jaunty ibex]

O_K is a dedekind domain

[junior veldt]

What does that mean?

[jaunty ibex]

if u know what

[raven plaza]

Monoid

[jaunty ibex]

nvm

[junior veldt]

I see

What’s the operation?

[raven plaza]

IJ = {ij}

[jaunty ibex]

I don't think it's very enlightening to say the definition of a dedekind domain

[junior veldt]

Ah, this is that multiplication of ideals definition

[jaunty ibex]

You have to see first why it's defined the way it is defined

[junior veldt]

Now here’s a confusion I have

As far as I know the definition of UFD applies to rings

Are you saying there’s a definition for monoids too?

[jaunty ibex]

Let me look up wtf was a monoid again

[junior veldt]

Just a set with an associative unital binary operation

[surreal sapphire]

a monoid is almost a group

you turn those guys into a group by formally adding fractions

[junior veldt]

Sure sure

[jaunty ibex]

But just associative and has identity right

[junior veldt]

Yeah

[jaunty ibex]

well you basically need to upgrade a monoid to a ring

[surreal sapphire]

this is the ideal class group which measure the failure of unique factorization in those rings of integers

[raven plaza]

I think one can consider strict unique factorizations in this setting

[surreal sapphire]

well, you have to quotient somewhere

[junior veldt]

What does this mean?

[raven plaza]

Since the only possible unit is the identity

[surreal sapphire]

but yeah, really factorization is a notion of monoids

it comes from multiplication

[junior veldt]

Like what I am confused by here is

Ok sure, the original ring is not a UFD

So instead you use ideal multiplication

[jaunty ibex]

yeah

[junior veldt]

But if the ideals don’t have a ring structure

Then it’s not like the ideals are a UFD

[jaunty ibex]

UFD implies dedekind domain but not the other way around

[junior veldt]

You’re misunderstanding

[surreal sapphire]

ye thats correct

[junior veldt]

As far as I can tell, the set of ideals doesn’t form a ring

[raven plaza]

It's a monoid

[junior veldt]

So to me it doesn’t even make sense to ask whether the set of ideals is a UFD or not

[wise bloom]

After so long

[surreal sapphire]

ye thats correct

they still factor uniquely

and you call the rings with this property (where ideals factor uniquely into prime ideals) dedekind domains

[wise bloom]

Hello guys

[jaunty ibex]

yea and that's the point

[junior veldt]

Is this in the monoid sense

[surreal sapphire]

sure

[raven plaza]

The UFD sense only uses the multiplication operation in its definition

[junior veldt]

Ok now I am understanding

[surreal sapphire]

each ideal can be written as a finite product of prime ideals

[raven plaza]

So it's the same

[junior veldt]

Yeah I was thinking

[surreal sapphire]

uniquely up to order

[junior veldt]

Cool this makes sense now

[jaunty ibex]

Somehow this makes less sense but sure

[junior veldt]

Ah, so I assume Z[alpha] for alpha an algebraic integer is a dedekind domain?

[surreal sapphire]

there might be a way to turn those into rings though

dedekind didnt know what ideals are

he called them ideal numbers

[wise bloom]

Hello

[surreal sapphire]

and there was a wholly different approach to this problem as well which is now forgotten

but probably nvm that 😄

[junior veldt]

So it makes to ask whether a monoid has the unique factorisation property or not

And for a UFD you’re just considering the ring as a monoid under multiplication

[jaunty ibex]

The thing is just because you have Q(alpha), Q(alpha) intersect Z is not necessarily Z[alpha] but yeah Q(alpha) intersect Z is a dedekind domain

[junior veldt]

But you can also take the ideals under multiplication as a monoid

[surreal sapphire]

no

only the rings of integers are

[junior veldt]

Ah

[surreal sapphire]

they are maximal orders

[junior veldt]

What’s a ring of integers

[surreal sapphire]

the non-maximal ones are not dedekind

the set of all algebaic integers over a number field

[junior veldt]

What’s a number field..

[surreal sapphire]

emphasis on all

finite algebraic extension of Q

[junior veldt]

Gotcha

So things which are roots of monic polynomials with coefficients in the number field…?

[jaunty ibex]

integer coefficients

[junior veldt]

I’m getting confused again

[jaunty ibex]

Nothing special just take the monic polynomial with integer coefficients

[surreal sapphire]

you take monic polynomials with integer coefficients

you can think of all of them over C if you want

[junior veldt]

What is an integer here

[surreal sapphire]

Z

[junior veldt]

Huh…

[surreal sapphire]

those are also called algebraic integers

[junior veldt]

Doesn’t this just give all algebraic integers

[surreal sapphire]

and for number fields you intersect those with your number field

ye, but intersect with your number field

[jaunty ibex]

literally x^n+c_1 x^(n-1)+... where c_1,c_2,... are integers

[junior veldt]

Hm…

Really trying to understand this

So the collection of algebraic integers is fixed

[jaunty ibex]

it's okay it's not easy

[junior veldt]

It’s a fixed subset of C

[jaunty ibex]

yes

[surreal sapphire]

like sqrt(3) doesnt live in Q(sqrt(2)) and thus not in its ring of integers

[junior veldt]

A number field is a field extension of Q that’s algebraic and finite

Ah ok hang on

So the “ring of integers” is a construction that takes in a number field and spits out a ring?

[shell vigil]

[surreal sapphire]

yes

[jaunty ibex]

yupie

[junior veldt]

That makes significantly more sense

[surreal sapphire]

its analogue of Z \subset Q but for Q(something)

[junior veldt]

Sure sure

And indeed the ring of integers of Q would be Z

In this sense

[jaunty ibex]

yep

[wise bloom]

Guys anyone interested in like practising or solving maa putnam or imo pyqs?

[jaunty ibex]

No

[junior veldt]

Ah ok so

Z[i] is the ring of integers for Q(i)?

[jaunty ibex]

Yes

[junior veldt]

And it just so happens that it can be expressed as “Z adjoin an algebraic integer”

[jaunty ibex]

Exactly

[junior veldt]

But in general rings of integers need not take that form

[jaunty ibex]

I can give you an example where this is not the case

Or maybe think about it

[junior veldt]

Sure!

Uh…

I doubt I’ll be able to come up with my own example

[jaunty ibex]

Q(sqrt(5))

Is it Z[sqrt(5)] or...

[surreal sapphire]

thats also of the form Z[algebraic integer]...

[jaunty ibex]

yeah

The point loch made was to take the maximal one

[junior veldt]

I’d have to use pen and paper I think

[jaunty ibex]

Sure!

Go ahead

[junior veldt]

Unfortunately I am outside rn lol

[jaunty ibex]

Oh okay

Well Z[sqrt(5)] is not enough

[vernal iron]

Time to boot up some random draw app on phone 💀

[junior veldt]

I see - so the ring of integers of Q(sqrt 5) is strictly larger than Z[sqrt 5]

[jaunty ibex]

Z[(1+sqrt(5))/2] turns out to be the full ring

[junior veldt]

Ah, because of the golden ratio?

Right!

[jaunty ibex]

yupie!

Deltoid kinda spoiled it

goddammit deltoid

[junior veldt]

Though yeah

Are there any rings of integers not of that form?

[surreal sapphire]

you may need to adjoin more

but they are always finite Z modules

[junior veldt]

I see I see

Right, so adjoining one only gives you a Z module of rank 2

But if I did, say

The ring of integers of Q(sqrt 2, sqrt 3)

I assume this does not take the form of Z[algebraic integer]

[jaunty ibex]

I would assume it is Z[sqrt(2),sqrt(3)]

[junior veldt]

That’s my assumption too but I’d have to check

[jaunty ibex]

But this is kinda getting out of topic

it's better to adjoint one thing

because sometimes you might get a field extension that's not Galois and lose some info

[junior veldt]

Idk what that means

But um

Thank you both

It feels like I actually understood something new in algebra

[jaunty ibex]

np

I'm glad tbh

[junior veldt]

This is very rare for me

[jaunty ibex]

Maybe NT might be ur thing

Okay I know what O_K could be

[junior veldt]

Haha, likely not

I’m a physicist, not a mathematician :P

[jaunty ibex]

An integral basis for Q(sqrt(2),sqrt(3)) is ||1, sqrt(2),sqrt(3),(sqrt(2)+sqrt(6))/2||

Woops

There

[narrow carbon]

vero with alg NT

[jaunty ibex]

I love alg NT

Literally the best field ever

[junior veldt]

Is this what alg NT is?

[jaunty ibex]

Yes

It feels like chemistry

[junior veldt]

It actually seems nice

[jaunty ibex]

Sometimes

[junior veldt]

Oh?

[jaunty ibex]

Yeah with the classifications

idk that's how I think of algebra sometimes

[junior veldt]

Again very rare for me to say this about anything that has “algebraic” in the name

[narrow carbon]

i dont exactly know what alg NT looks like, but i know vero always talk about alg NT

[junior veldt]

could you elaborate?

[jaunty ibex]

classifications

in alg NT for example you classify O_K's

It's kinda like in chemistry you have a periodic table

[junior veldt]

Can you remind me what O_K is?😅

[jaunty ibex]

Ring of integers :3

[junior veldt]

Mhm, I’m aware

[junior veldt]

Ahhhh I see

[jaunty ibex]

I forgot why we use O for ring

[junior veldt]

roing

[deep mango]

normal people sorting stuff: ugh i cant wait to be done with this
algebraists sorting stuff: UNNFFF IM SOORTIIIIIING!!!!!

[junior veldt]

[junior veldt]

this might sound dumb but I haven’t really thought of algebra that way before

[jaunty ibex]

It's okay lol

[vernal iron]

Clearly cause 💍 shape looks like O 💀

[junior veldt]

When I think algebra I think symbol-pushing

[jaunty ibex]

My advisor doesn't think of algebra this way

[junior veldt]

But classifications actually seem fun

[jaunty ibex]

Clever

[deep mango]

sheaf

[jaunty ibex]

I know I use A for rings

[surreal sapphire]

order

[jaunty ibex]

Because french

[jaunty ibex]

Ordung?

[surreal sapphire]

Ordnung, yes

[jaunty ibex]

oh thanks

[vernal iron]

Makes sense

[surreal sapphire]

but in english its order

the more general theory is that of orders inside a number field

of which the ring of integers is the maximal one

[raven plaza]

What's an order

[junior veldt]

Unironically this seems cool

[jaunty ibex]

My prof was german and I faintly remember he said something similar to "ordung" and there we go

[junior veldt]

Is that weird to say

[deep mango]

its not weird in that we all expected it

[jaunty ibex]

No lol

[surreal sapphire]

the definition of order is a bit complicated

[junior veldt]

Wdym

[junior veldt]

I am just very not used to liking or understanding algebra

[raven plaza]

Is it related to valuations

[jaunty ibex]

Makes sense

I mainly like algebra cuz idk I just like it ig

[vernal iron]

Makes sense

[jaunty ibex]

it doesn't feel very cancerous like analysis

[surreal sapphire]

no

the word order is very overloaded

[vernal iron]

Lol true

[junior veldt]

Ah in my case I love analysis

[jaunty ibex]

Order outside math vs order in math:

[junior veldt]

But if I think of algebra as the maths of organising/sorting/classifying

Then I think I could actually like it a lot

[vernal iron]

Idek what order means outside of math anymore, like what, not chaos?

[raven plaza]

It is

[gusty socket]

lol

[junior veldt]

What is

[vernal iron]

Maybe it is no sleep brain

[raven plaza]

I think of algebra as organizing / sorting data

[junior veldt]

Wait fr

[gusty socket]

data

[raven plaza]

Well I also consider all of cats to be algebra

[vernal iron]

datum would be if you only had one thing, data would be better here

[junior veldt]

I think I actually understand why people say that now

[jaunty ibex]

I like geometry as well but I am incredibly awful at making geometric arguments

[raven plaza]

Datums

[vernal iron]

So true

[jaunty ibex]

So I naturally just chose algebra

[junior veldt]

In my case I love making geometric arguments

[junior veldt]

Because I love the idea of organising or sorting stuff

[raven plaza]

This is solely why I like algebra

[junior veldt]

whaaaaaaat

i…

i didn’t realise

[surreal sapphire]

but uhh in the number field case an order is a subring that Q-spans the number field and is a finitely generated Z-module

[vernal iron]

My autistic ass brain when it comes to sorting:

[raven plaza]

I am a true fsct at heart

[surreal sapphire]

maybe you need something else i forgor

[junior veldt]

I literally used to carry out mergesort on a deck of cards for fun

[vernal iron]

💀 crazy work

[junior veldt]

Does this mean I am allowed to like algebra now

[jaunty ibex]

Maybe physics majors will find jobs before cs majors

[raven plaza]

Welcome to the gang

[vernal iron]

You're allowed to like whatever you want

[vernal iron]

:pain:

[gusty socket]

you were always allowed to like algebra

[raven plaza]

No you aren't, it is a fate decided by the math gods

[jaunty ibex]

@latent edge we got pseudonium understanding the stuff btw

[junior veldt]

This is quite a lot for me to take in

[jaunty ibex]

so dont feel bad

[vernal iron]

There's no way he isn't asleep rn

[junior veldt]

I have been afraid of algebra for, like, years

[latent edge]

Incredibly based

[vernal iron]

No way bro is awake

Crazy work

[junior veldt]

hi deltoid

[shell vigil]

LMAO

Bro was just waiting

[latent edge]

i was reading automorphic forms lol

[junior veldt]

and I haven’t allowed myself to like it during that time

[latent edge]

Sleep is cooked anyway

[jaunty ibex]

bro is drunk af

[vernal iron]

Sleep is cooked yes

[junior veldt]

I think it might take a while but this might genuinely help me overcome my fear of algebra

[umbral thorn]

Algebra is veryyyyyyy easy if u rinterested

[jaunty ibex]

One step forward

[junior veldt]

Genuinely thank you so much

[vernal iron]

Time to lock in on algebra

[raven plaza]

Pseudo has so much trauma from Cambridge

[jaunty ibex]

Doesn't pseudonium know quite a good chunk of category theory

[junior veldt]

Mhm…

[jaunty ibex]

This is already good enough for algebra pilled arc

[junior veldt]

Basic category theory, yeah

[jaunty ibex]

May I ask what happened

[junior veldt]

#math-discussion message

[jaunty ibex]

Yeah nG is right

For example

I don't find modular forms to be elegant

I do find number fields to be elegant

Oh I see

I'm sorry to hear this

[latent edge]

I still don't believe u

[jaunty ibex]

Incoming 1000000 words long modular forms yappery:

[vernal iron]

True

[jaunty ibex]

Idk modular forms have two sides I dislike

Lie theory and harmonic analysis

[narrow wraith]

Guys can multi variable calculus be learnt in 3 days for an exam?

Asking for a friend

[vernal iron]

Probably

[jaunty ibex]

@narrow wraith

Memorize this pic

[vernal iron]

Real

[jaunty ibex]

This too

[vernal iron]

Real part 2

[narrow wraith]

Dang I don’t understand any of that

[jaunty ibex]

The first picture already gives calculus 3 + diff geo

[vernal iron]

Maybe not learnt well

But learnt yes

Gl

[narrow wraith]

Enough to get a 6

On the exam

Thank u

[vernal iron]

Idk what a 6 is

Bur sure

[shell vigil]

What's a 6 lol

[narrow wraith]

6/10

[vernal iron]

Yes probably

[narrow wraith]

Oki

[vernal iron]

Start studying now

[narrow wraith]

Yee I started 2 days ago

Just thinking about the worst case scenario

[vernal iron]

No sleep until night before in which case 8 hours at least

[narrow wraith]

Test on Monday

Oki

[jaunty ibex]

This picture shows cochain complexes

Cohomology...

[vernal iron]

Joking

Sleep is important

[shell vigil]

Cocaine complexes

[narrow wraith]

😂🙏🙏

[vernal iron]

Just saying you can do like 6 hours of sleep until night before in which case 8 at least

[raven plaza]

6 is the dual of ∂

[jaunty ibex]

Cohomology is a quotient on cocaine complexes yes

[narrow wraith]

x∂

[warm umbra]

It's a whole new world in here.

So many familiar faces.

So many possible adventures.

We truly are in the greatest moment.

[vernal iron]

This channel is better than discussy

[jaunty ibex]

I don't know you respectfully

[warm umbra]

Cherish it.

[warm umbra]

https://tenor.com/view/crying-emoji-gif-21922016

[vernal iron]

Side note it is still bad

[warm umbra]

lmao

[umbral thorn]

@jaunty ibex come dm

[jaunty ibex]

I'm taken. Thanks

[vernal iron]

Crazy work

[warm umbra]

welcome to the server @umbral thorn

mf tryna rizz in mathcord 🙏

[deep mango]

@raven plaza come dm

[vernal iron]

Crazy work part 2

[jaunty ibex]

@vernal iron come dm 😉

[vernal iron]

💀

Crap I cannot send images here

Need to be more active smh

[jaunty ibex]

Lol weak

[vernal iron]

True

[jaunty ibex]

@latent edge come dm

[deep mango]

oh heavens

[latent edge]

I'm good

[long matrix]

dick move by ryc

[umbral thorn]

@latent edge come dm'

[warm umbra]

@umbral thorn come dm

[umbral thorn]

ya sure

[long matrix]

[shell vigil]

Bros gonna send u a pyramid scheme link

[vernal iron]

Lol

At least it isn't a pyramid stack

[shell vigil]

I'm having lunch at 5 pm why do i even bother

[deep mango]

girl? dms now.

[jaunty ibex]

Stack? Algebraic geometry?

boy? dms now.

[shell vigil]

Just to tell him "I'm taken"

[vernal iron]

Lol yes

[vernal iron]

I DMed her and she said that to me lol

[jaunty ibex]

What have he done to u

[vernal iron]

Deltoid? Mentioned them in passing out of fear

[shell vigil]

LMFAO leaving no chances

[jaunty ibex]

I feel like Deltoid probably gave you a distorted view of what math to do

[vernal iron]

Definitely

[jaunty ibex]

[warm umbra]

It's been a journey.

[jaunty ibex]

Beyond cooked

[vernal iron]

Yeo

Yep

[warm umbra]

But we all grew as people.

[shell vigil]

You're a father of three you definitely grew more than any

[warm umbra]

Even in the lows; even in the highs, it was all necessary for character.

[warm umbra]

[jaunty ibex]

Wtf is bludwin yappin about

[vernal iron]

The worst part is that I find it interesting lol

[jaunty ibex]

Good lol

[warm umbra]

What can I say; i'm one of a kind.

[shell vigil]

No there's a million more blond penguins out there

All preaching peace

[warm umbra]

https://tenor.com/view/pingu-rising-from-water-hello-hi-waving-gif-5193317710960738776

[jaunty ibex]

Bro says that and he barely came out of his mom's womb

[warm umbra]

[shell vigil]

One of a kind in a different way than he intended

[warm umbra]

Exactly.

I'm the only Yeet.

[vernal iron]

Unfortunately no

[shell vigil]

Can anyone gift me nitro

[warm umbra]

@carmine flower It's great to finnaly meet you.

[vernal iron]

Why do you want nitro

[shell vigil]

Just for a year nothing much

[shell vigil]

Cause im boutta lose it and i wont be able to use menhera anymore

Menhera emojis**

[vernal iron]

Just send gifs smh

[shell vigil]

Take up too much space

[vernal iron]

Can I dm you rq

[shell vigil]

K, but send me nitro as well lmao

[vernal iron]

Chat thinks I am gonna be a sugar momma

[shell vigil]

Will you

Nah dont spend shi on nitro

Buy food instead

[jaunty ibex]

Buy food for me. Nothing for u

[long matrix]

i dont think begging aligns with the rules

[shell vigil]

Does everybody just ask to dm me cause i have ask on server in my name and then just never dm me

[vernal iron]

Crazy work

[vernal iron]

I ask everyone

[shell vigil]

And do you ever dm

Or just

Say that

[vernal iron]

I just did dm

[shell vigil]

O

[jaunty ibex]

Bro's explaining a conspiracy theory rn

[vernal iron]

Who

[jaunty ibex]

U

[warm umbra]

the theory is crazy btw

they sent it to me too

[jaunty ibex]

Ur mom is the theory

[vernal iron]

Conspiracy theory?

Huh?

Y'all I can tell you a conspiracy theory later if you want

But I haven't talked about any recently

[warm umbra]

💔

[vernal iron]

Also if possible could you not use "bro" for me anymore (I was fine with it in the past but like /shrug Idk)

[jaunty ibex]

Oh yeah totally forgot

Thanks for reminding

[vernal iron]

Nah you're all good

[umbral thorn]

nonesense

[warm umbra]

hey

[umbral thorn]

?

[latent edge]

Meow

[vernal iron]

Meow indeed

[elfin smelt]

mrow

[shell vigil]

Meow

[vernal iron]

Chat I become active in this server for 1 night and I get dms from 2 incels

[shell vigil]

Lmfao, expected tbh

[vernal iron]

It's that bad?

Now I feel bad for pre-transition me's actions (I was never an incel but like)

[visual breach]

Imagine DMing in discord like that

[vernal iron]

true

[shell vigil]

I mean there's outwardly thirsty people irl, so it's guaranteed there'll be 10X online

[vernal iron]

true

[rancid tendon]

bruh you couldve asked like 1 month ago when i could still share my nitro

o wait lemme read all the messages i mightve missed

[shell vigil]

Lmaos

[fresh comet]

@umbral thorn welcome to the mathcord!

[ocean harbor]

you already have one tho

[shell vigil]

It runs out in 6 days

[ocean harbor]

sad

you can maybe claim one day nitro from quest

to make 7 days

[shell vigil]

Lmfao I ain't gonna install a game and play it for a day of nitro

[ocean harbor]

its sometimes video

[shell vigil]

Might do that

But definitely no games

[ocean harbor]

yeah i never do games either

[shell vigil]

Too much of a bother tbh

[ocean harbor]

real

[shell vigil]

Then they bug you with promotions and whatnot

The more places I give my email address away, the more shi i get

[ocean harbor]

im still stuck on one year nitro

well i still have one month more

[solid snow]

@fresh comet https://www.youtube.com/watch?v=hiDq5ECnx7c

[ocean harbor]

so close

[fresh comet]

[solid snow]

these are hilarious

[shell vigil]

You bought it?

Or gifted?

[ocean harbor]

gifted ye

[shell vigil]

Pogs

[ocean harbor]

fr

[solid snow]

ah wait

https://www.youtube.com/watch?v=ePbMNq8m7Jw

this one is especially funny

[shell vigil]

I might just buy the basic version for a month more if i go through too much withdrawal

Atleast i get emojis with that

Though even that's overpriced

[ocean harbor]

nitro basic used to be cool before but discord changed it

you could use animated avatar before

L discord

[shell vigil]

Exactly, emojis and avatar are the only two things I need out of nitro, the rest is all trash, and dc still took one of those away

[ocean harbor]

[jagged forge]

jordan klepper pretending to be a conservative is just such a good bit for some reason

man has a talent

[solid snow]

i love the way he teases out contradictions and just... dumb responses by not even really doing anything

[barren kelp]

mhm

[jaunty ibex]

Lol normal

Wait till u get creeps asking for face reveal

[old oak]

I never got why some men find it so hard not to be a creep.

I'm sorry you have to put up with that

[floral lake]

hi

[vernal iron]

😔

[jaunty ibex]

@latent edge have you read about opetopic omega categories