Mathematics

Akiva Weinberger

7:00 PM

Did we define poset

It's a bunch of things with a "less than" relation except sometimes we neither have $a\prec b$ nor $b\prec a$ nor $a=b$

Example: Sets, using $\subset$ as our order relation

A maximum element is $\succ$ everything else. A maximal element is not $\prec$ anything.

Partially ordered set

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total...

Adeek

@AkivaWeinberger do you want to discuss something ?

I want to discuss this proof here

for the first thing the result is true for n - 1 the reason we require whenever $j \neq i$ because if $j = i$ then we are done by hypothesis right ?

Akiva Weinberger

Remind me what a prime ideal is?

Generated by one element?

SteamyRoot

If you have a set $X$ with a minimum and maximum.

Consider $\varnothing \subseteq X$.

Adeek

@AkivaWeinberger an ideal is prime if whenever $ab \in I$ $a \in I$ or $b \in I$.

SteamyRoot

Then $\inf \varnothing$ is the maximum and $\sup \varnothing$ is the minimum.

Akiva Weinberger

7:05 PM

Yup @SteamyRoot

Adeek

So it is really like prime numbers.

SteamyRoot

Which is one of the reasons it's terribly important to exclude the empty set (if necessary) when working with subsets :D

Adeek

@AkivaWeinberger okay ?

Akiva Weinberger

Finished reading it

Yeah, the first bit with the $i\ne j$ follows from the induction hypothesis

Adeek

Then why does $y \notin p_i$?

I don't understand the concluding remark.

Akiva Weinberger

7:10 PM

Because all of the terms except for one are in $p_i$, and the last one isn't.

The "last one" is the one with $x_i$ omitted.

Because all of its factors are not in $p_i$ by the way we chose our $x$s.

Forever Mozart

i just fried up some salt pork and it doesnt taste good

I thought it would be like bacon, but ugh

Adeek

oh ok I see it now. The way they phrased the last statement caught me off guard. that is $x_i \in p_i$ for all i.

but yeah here we have $x_i$ missing so that makes sense ok yeah thanks @AkivaWeinberger

what's nu

$c/\lambda$

nah that's $n$

7:20 PM

Hi chat

Adeek

I guess this makes also intuitively this proposition.

I guess what this proposition says is that ideals can't spread over many different ideals.

It must be contained in one of them, which makes sense intuitively.

Alessandro Codenotti

Hi @Astyx

Tobias Kildetoft

@Adeek Also note the similarity between this and the condition to be a prime ideal (if you write this result for two prime ideals)

Adeek

Yeah @TobiasKildetoft

I guess this condition captures intuitively what it means to be prime ideal more.

Extrarius

How can an endomorphic ring be isomorphic to a quadratic order?

Socrates

7:33 PM

can you fix something when you don't know what's broken?

appereantly yes, format c and install the OS again

Tobias Kildetoft

@Extrarius what does it mean to be endomorphic?

Akiva Weinberger

Test: $\displaystyle\operatorname*{\bigtriangleup}\limits_{i=1}^3A_i$

Extrarius

@TobiasKildetoft err I meant endomorphism ring

Tobias Kildetoft

@Extrarius And by quadratic order, you mean an order in a quadratic number field?

Extrarius

@TobiasKildetoft The paper I'm reading says "Let $\mathcal{O}$ be an imaginary quadratic order identified by its discriminant D". A few sentences later it refers to the same object as "the lattice $\mathcal{O}$"

Akiva Weinberger

7:39 PM

Test: $\displaystyle\operatorname*{sum}\limits_{i=1}^n\operatorname*{prod}_{\substack{‌1\,{\rm le}\,j\,{\rm le}\,n\\i\,{\rm ne}\,j}}\frac{x-x_j}{x_i-x_j}$

Tobias Kildetoft

@Extrarius Right, an order is a maximal lattice (i.e. maximal free $\mathbb{Z}$-subalgebra)

Extrarius

I'm not familiar enough with group theory to understand that. I just recently wrapped my head around an endomorphism ring being a "function rings" with composition for multiplication.

Tobias Kildetoft

@Extrarius In that case I think reading a paper on algebraic number theory is getting way ahead of yourself

Akiva Weinberger

Wait, does ANT refer to algebraic number theory or analytic number theory

Tobias Kildetoft

@AkivaWeinberger Depends on who says it probably

Alessandro Codenotti

7:44 PM

a small insect when said by most people

Akiva Weinberger

That's ant

ANT is much bigger

$\huge{ANT}$

Extrarius

@TobiasKildetoft It is worse than that, I'm reading a paper on accelerating the CM method for constructing elliptic curves =-) What titles/labels should I search for to find material that can help me understand orders and lattices and such

Tobias Kildetoft

@Extrarius I would start with some basic algebra, so rings and group and such. There is a long way to go to the topics used in papers

Balarka Sen

@AkivaWeinberger what about TNT

transcendental number theory

Extrarius

@TobiasKildetoft: When you say basic algebra, would that be abstract algebra? It makes me think of solving simple equations and polynomials etc, but I'm guessing that isn't what you mean

Akiva Weinberger

7:48 PM

@BalarkaSen Don't give it to the ants.

Tobias Kildetoft

@Extrarius Right, abstract. I never add that word because it should be implied from the context of university math

Extrarius

@TobiasKildetoft Do you know of any good books or other resources on the topic?

Tobias Kildetoft

@Extrarius I usually recommend Artin since it is often recommended by others. Haven't read it myself though

There are tons of good ones to pick from

Extrarius

@TobiasKildetoft When I search for author:Artin on amazon, I get a LOT of books by a few different artins (Michael and Emil, mostly). Any idea which one(s)?

Tobias Kildetoft

@Extrarius Michael

Book is simply called "Algebra"

not sure which is the newest edition

Extrarius

7:55 PM

@TobiasKildetoft Thanks. I wish self-education was less expensive =-(

Tobias Kildetoft

Ahh, there are probably also free books that are good. Let me take a look and see

Balarka Sen

artin can be downloaded

eg from libgen

Tobias Kildetoft

8:07 PM

abstract.ups.edu and math.uiuc.edu/~r-ash/Algebra.html both look like pretty decent free options. The latter even has a chapter on algebraic number theory

Extrarius

@TobiasKildetoft Thanks

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$Mathematics$ Mathematics