Mathematics - 2014-04-09 (page 1 of 4)

Pedro Tamaroff

00:02

@KarlKronenfeld I was eating. But now I feel better and might think about something.

Just some minutes.

Ted Shifrin

Nutrition is crucial for math. :)

Pedro Tamaroff

@KarlKronenfeld What about... $(2)$ and $(X^2+2)$?

I was literally feeling sick.

Now I feel amazing.

Karl Kronenfeld

try it :p

Pedro Tamaroff

SIGH.

Fernando Martin

Hi @Ted

Pedro Tamaroff

00:05

$(2,X^2+2)$ is prime.

Right?

Karl Kronenfeld

yup

Ted Shifrin

It is?

Pedro Tamaroff

@TedShifrin Yes.

Fernando Martin

x^2 is irreducible over Z/2Z

Karl Kronenfeld

It's maximal, in fact, @TedShifrin

Fernando Martin

00:05

so yes

Jasper Loy

@TedShifrin Notation is crucial for math

Fernando Martin

wait

Ted Shifrin

The quotient isn't an integral domain!

Karl Kronenfeld

you're right, where am I going wrong?

Pedro Tamaroff

But $2$ is prime and $X^2+2$ is irreducible.

Fernando Martin

00:07

@Pedro: you need it to be irreducible over Z/2Z

Ted Shifrin

Y'all be wrong.

Karl Kronenfeld

Ah, now I see my misconception

Fernando Martin

In general $(p, f)$ is prime if $p$ is prime and $f$ is irreducible mod p

Pedro Tamaroff

@FernandoMartin Oh, $\mod p$.

Ted Shifrin

Agreed @Fernando.

Pedro Tamaroff

00:08

Then I think I am right.

Jasper Loy

@TedShifrin I keep quiet, so I cannot be wrong

5

Fernando Martin

but x^2+2 = x^2 isn't irreducible

Pedro Tamaroff

Thinks.

Ted Shifrin

Clever @Jasper. I don't mind being wrong :)

Pedro Tamaroff

@FernandoMartin I was saying $(2,X^2+2)$ contains $X^2$ but doesn't contain $X$. So it is not radical.

I was right all along.

Fernando Martin

00:10

Oh, I thought you were trying to prove it was(n't) prime

Pedro Tamaroff

@FernandoMartin Nope.

I blame @Karl >:(

Karl Kronenfeld

I did crap like $(2x,x(x-1))$.

Jasper Loy

@PedroTamaroff Wow, that face sure looks angry

user127001

I need to buy an air conditioner for my room

Fernando Martin

@Pedro: is it me or fb is loading reaaaaaaally slow today?

or not even loading at all

Jasper Loy

00:12

@user127001 Which other math courses have you failed?

Pedro Tamaroff

@FernandoMartin Loaded for me.

user127001

@JasperLoy just Algebra

Ted Shifrin

I had no probs @Karl

user127001

I have As in everything else but I've only taken easy courses besides Algebra

Jasper Loy

@user127001 Does the instructor use a textbook for that course?

user127001

00:13

@JasperLoy no

Pedro Tamaroff

►

Jasper Loy

@user127001 I think finding a text that suits you will help

user127001

He goes in his own order

The first day he went over Lagrange, Cauchy, and Sylow's theorems

Then he introduced groups later

Pedro Tamaroff

@user127001 That's ridiculous.

Ted Shifrin

Holy crap @user127001

Pedro Tamaroff

00:15

Unless time-space is reversed in your uni.

Jasper Loy

Gee, what order is that?

Karl Kronenfeld

@user127001 how?

user127001

But he can do whatever he wants because he's a world famous expert on group theory

Pedro Tamaroff

@user127001 No, he can't.

Ted Shifrin

Hmm, who is he?

user127001

00:15

John H. Conway

Jasper Loy

@user127001 What is that course about again?

Ted Shifrin

Oh wow .... At Princeton?

user127001

Algebraic structures

Groups, rings, fields, etc.

No @TedShifrin he's taking a break from Princeton and teaching at our school for this semester

Ted Shifrin

Checks middle initials ....

Jasper Loy

@user127001 What school is that?

user127001

00:17

He's the game theorist and surreal numbers guy

@JasperLoy it's a local city college here

I don't think we are even listed on USNews

Ted Shifrin

Wow ... He's a superstar in all sorts of math ... But nothing if not idiosyncratic.

Jasper Loy

@user127001 I did mu undergrad in a place with good rankings but a poor curriculum lol

user127001

We have poor rankings and poor curriculum

Ted Shifrin

I think I'm the only person retiring before 80 :)

Jasper Loy

@TedShifrin I intend to work till death lol

@user127001 Wow that sounds bad

Studentmath

00:19

"rest in the grave"

Ted Shifrin

So, where are you? @user127001

Jasper Loy

@TedShifrin He is in chat haha

user127001

@TedShifrin I am in NY

Jasper Loy

The worst grade I got for a math course was B+

Ted Shifrin

I meant which college, silly boy ...

user127001

00:22

Oh, Queens College

Ted Shifrin

I got a B in college in the course which is now my expertise ...

Studentmath

The Irony

Ted Shifrin

@user127001: Far from an unknown college!

Pedro Tamaroff

@FernandoMartin

You doing any AM?

Jasper Loy

I wasted 20,000 bucks for a shitty undergrad education

Fernando Martin

00:23

not right now

user127001

@TedShifrin you know of it?

Jasper Loy

I want a refund lol

user127001

I got into some other schools but could not afford it

So I went with the school that gave me full scholarship

Jasper Loy

@user127001 Well just get into a better grad school

Ted Shifrin

Yes, indeed, @user127001. BTW, it seems Conway retired from Princeton.

Jasper Loy

00:24

@user127001 Not a bad idea

Pedro Tamaroff

@KarlKronenfeld The obvious Zorn proof doesn't work to show that a commutative unital ring contains prime ideals minimal wrt to inclusion.

user127001

Oh, so I guess he decided to mess about with some lower class schools

Jasper Loy

@user127001 Hope you change your name soon

user127001

@JasperLoy I think I should have taken on the debt for a school with better pedagogy in their professors

Studentmath

Wow, it's really an economic burden to study in the US

Jasper Loy

00:28

I am not from US

Studentmath

Where from then? (if I may ask)

Ted Shifrin

Unless it's a famous liberal arts school known for math, you don't necessarily get great teaching for the money.

Jasper Loy

@Studentmath Singapore

Studentmath

Ahh

Well, it's really an economic burden to study in Singapore too, I guess

Jasper Loy

Well, the education I got was def not worth the money lol

Studentmath

00:31

Oh well, actually it's not that cheaper in here. State-recognized university will cost you about 15k for B.Sc

And the taxes, oh god the taxes..

Ted Shifrin

Where are you? @Studentmath

Studentmath

Israel

Ted Shifrin

Ah, cool.

user127001

@TedShifrin Coincidentally someone at school just recommended me your diff. geo. book as a supplement to vector calc...what..

Ted Shifrin

See, @user127001, free bad teaching :)

skullpatrol

00:32

Hi Professor @Ted

Ted Shifrin

Diff geo is for when you already know vector calc and linear alg

Hi @skull

user127001

The pre-requisites for our diff. geo. course is only vector calc and lin. alg., is that literally all you need?

Ted Shifrin

That's our prereq too ... Some experience with proofs is good.

Studentmath

Prof. @Ted , you have a textbook about multivar. Calculus and Linear Algebra, correct?

user127001

What is a good book to learn how to do proofs well?

Ted Shifrin

00:35

Yes @Studentmath ... Integrated together.

Best is in context with lin alg or alg, @user127001, but there are some books to help approach proofs. I like a book by Kevin Houston.

skullpatrol

So it's best to do it the way the prof does it ;-)

Studentmath

I shouldn't have taken these computer science courses..

user127001

@TedShifrin is it "How to Think Like a Mathematician"

Ted Shifrin

Yes @user127001

user127001

Thank you

skullpatrol

00:43

How many people are in your modern algebra class @user127001?

user127001

I think less than 10

skullpatrol

Just to help me get an idea of the competition

user127001

I know one of them teaches remedial math at our school

Another guy quit his job in his 30s to come back to get a masters in math

I don't know about anyone else

I just overheard those 2 talking

Ted Shifrin

This is a grad course or an undergrad?

user127001

Undergrad

Ted Shifrin

00:46

Crazy

skullpatrol

Why crazy?

Ted Shifrin

See what @user127001 posted 20 minutes ago or so

skullpatrol

Yes sir.

Crazy indeed prof @Ted

Pedro Tamaroff

@anon

Jasper Loy

I don't see anon @PedroTamaroff

Ted Shifrin

00:54

Is @Pedro hallucinating?

Pedro Tamaroff

@TedShifrin He's around.

Jasper Loy

He is around; I am round, lol.

Ted Shifrin

No hyperbole, @Jasper

skullpatrol

Hyperbolic speakers are kindly directed to the English language & usage room :-)

Ted Shifrin

Only elliptic and parabolic speakers here?

user127001

01:10

I got a $500 Barnes & Noble gift card

I'm afraid to buy textbooks though because I don't have anywhere to hide them

I might just re-sell the card

Mike

01:22

I laughed @Ted

Did he leave again? I always show up 10 minutes after he does.

Pedro Tamaroff

@Mike That's because you play shameful videogames.

This is your punishment.

eXtremiity

!!!!!

My assignment is to replicate the 2048 game!

Programming is so great =D !

Mike

I categorically deny all of your accusations, @Pedro

Pedro Tamaroff

@Mike Will you draw some diagrams to refute me?

Studentmath

Take it back @eXtremiity !

eXtremiity

01:31

Hahaha !

Karl Kronenfeld

@PedroTamaroff The intersection of a chain of prime ideals is prime, though.

Pedro Tamaroff

@KarlKronenfeld Yes, I just proved that. =)

Karl Kronenfeld

Then, why does the obvious Zorn's lemma argument not work?

Mike

Because Pedro doesn't believe the axiom of choice.

Karl Kronenfeld

I wonder if anyone has attempted to do modern commutative algebra without the axiom of choice

Ted Shifrin

01:39

You laughed? @Mike

Pedro Tamaroff

@KarlKronenfeld Because I was being silly.

Ted Shifrin

Believe or believe in?

Karl Kronenfeld

@PedroTamaroff I just now realized that it may not have been the obvious choice for you. Were you trying to find a minimal nonzero ideal or something?

Pedro Tamaroff

@KarlKronenfeld Minimal prime ideal.

Mike

01:56

@Ted At 'parabolic or elliptic only"

anon

@PedroTamaroff for some reason your ping did not reach my inbox

Pedro Tamaroff

@anon Oh noes.

@anon I am about to say something very correct or very silly.

anon

or both

Pedro Tamaroff

Is $k[x,y]/(xy)$ even close to being isomorphic to $k[x]\times k[y]$?

anon

or neither

Mike

01:58

Instead you should say nothing at all

Pedro Tamaroff

it looks like something similar...

Mike

@pedro try checking the 'obvious' map

Pedro Tamaroff

My problem is where to put the independent coefficient.

Mike

from the second to the first

Pedro Tamaroff

Well, yes.

Mike

02:00

alternatively

Pedro Tamaroff

Any bivariate polynomial can be written as $c+p(x)+q(y)+xyr(x,y)$.

Karl Kronenfeld

that'll be a problem @PedroTamaroff, methinks

Mike

send $k[x,y]$ to $k[x] \times k[y]$, and check the keenel

Pedro Tamaroff

@KarlKronenfeld What will be?

@Mike Right, but where do you send the independent cofficient?

Karl Kronenfeld

@PedroTamaroff The constant doesn't have a canonical place, as you mentioned

Pedro Tamaroff

02:01

@KarlKronenfeld Yes.

That's my point.

Mike

ah, you're right

Karl Kronenfeld

How about the tensor product?

Mike

they won't be isomorphic

Pedro Tamaroff

So it is something like that, but not quite.

anon

let $L$ be the image of $k[x,y]/(xy)\to k[x]\times k[y]$. then $k[x]\times k[y]\cong k\times 0\oplus L\cong 0\times k\oplus L$ I think

Mike

02:02

tensor won't work methinks

Pedro Tamaroff

Wait.

@anon What is the map?

anon

$p(x,y)\mapsto (p(x,0),p(0,y))$

Pedro Tamaroff

@anon Ah, OK.

anon

and that's $(k\times0)\oplus L$ and $(0\times k)\oplus L$, as rings even I believe

Pedro Tamaroff

I was thinking the quotient is iso to $k\times (x)\times (y)$ with appropriate multiplication.

anon

02:06

what quotient?

Pedro Tamaroff

$k[x,y]/(xy)$

anon

what is the copy of $k$ sitting inside $k[x,y]/(xy)$?

(you mean as $k[x,y]$-modules right?)

both $(x)$ and $(y)$ are iso to $k[x,y]$ itself as a $k[x,y]$-module

- bleh -

Pedro Tamaroff

$$\eqalign{
& \left( {c,xp,yq} \right) \cdot \left( {c',xp',yq'} \right) = \left( {cc',c'xp + cxp' + xxp'p,c'yq + cyq' + {y^2}q'q} \right) \cr
& \cr} $$

anon

CRT doesn't apply since $1\not\in(x)+(y)$

what is that monstrosity?

Pedro Tamaroff

What I think multiplication should be like in $k\times (x)\times (y)$ to make it look like $k[x,y]/(xy)$.

That's how things multiply in the quotient.

Studentmath

02:11

@eXtremiity the fear of the infinite loops..

anon

oh, you mean an operation on the underlying set of $k\times(x)\times(y)$, not the direct product structure

then sure

(you should have just said that)

Pedro Tamaroff

So the one should be $(1,0,0)$.

@anon Sorry, yes. =)

anon

well, $k\times kx\times ky$

Karl Kronenfeld

but x^2 and whatnot are also members

Pedro Tamaroff

I have no idea how that should help to find the minimal primes in $k[x,y]/(xy)$.

eXtremiity

02:12

Ah, the infinite loops. Well, at least CNTRL+C handles that problem :) @Studentmath.

They are annoying. And arrays. Arrays seem to be quite stubborn.

Pedro Tamaroff

I just wanted to see what that quotient behaved liked.

Studentmath

God, I am taking a course soley about Arrays right now.

anon

@PedroTamaroff a minimal prime in $k[x,y]/(xy)$ should lattice correspond to a prime of $k[x,y]$ containing $(xy)$ with no intermediates

eXtremiity

I see where your pain may be stemming from.

Pedro Tamaroff

@anon Yes.

eXtremiity

02:13

Either way, first year programming is so much fun.

Studentmath

They have no trust in us, before pretty much every single algorithm they have in bolded: "Watch out from infinite loops!!!!"

Karl Kronenfeld

@PedroTamaroff It's much easier to just study $k[x,y]$, as anon said.

Though, for the harder ones, geometry becomes relevant.

Studentmath

Programming itself is real fun, yes. If math gives you this amazing feeling when you get to solve a problem, programming gets you the feeling as if you can create anything you want

Until you start working with Arrays, that's it

Mike

@Ted I didn't get to whine about homework to you.

Karl Kronenfeld

Why the hatred toward arrays @Studentmath

eXtremiity

02:19

Hahahah :D

Studentmath

Just joking @karl :) Very useful and all that. Finally get to use some of my probability and calculus and feel mathematically smarter and useful, plus graph theory. But building and programming them right.. it's a pain sometimes.

Pedro Tamaroff

@KarlKronenfeld Can you amaze me with a geometry relevant example? =)

Karl Kronenfeld

I won't give an amazing example, since we can use yours quite effectively. Geometry can help you locate the minimal primes, as there is a one-to-one correspondence between the irreducible components of a solution set to a (system of) polynomials and the minimal prime ideals containing the ideal generated by those polynomials.
If you graph the solutions to $xy=0$ in the plane, you see that it is the union of two lines, the solutions of $x=0$ and $y=0$. Then, your minimal primes will certainly be $(x),(y)$ if $k$ is algebraically closed. If $k$ is arbitrary, you at least know what to try.

Pedro Tamaroff

@KarlKronenfeld Where does $k$ being alg. closed hop in?

Karl Kronenfeld

@PedroTamaroff It is needed to correspond points (a,b) with maximal ideals (x-a,y-b)

Pedro Tamaroff

02:34

@KarlKronenfeld Ah, yes, our prof. mentioned that. But we haven't really talked about it yet.

If $\mathfrak p$ is any prime with $(xy)\subseteq \mathfrak p$ then $xy\in\mathfrak p$ so $x\in\mathfrak p$ or $y\in\mathfrak p$, that is, for any prime containing $(xy)$ either $(x)$ or $(y)$ are inside. So certainly this two are minimal.

And I have proven either one is always in the prime... so that's it isn't it? @KarlKronenfeld

Karl Kronenfeld

Yes.

Pedro Tamaroff

So here the point is that $\{xy=0\}=\{x=0\}\cup\{y=0\}$?

Karl Kronenfeld

@PedroTamaroff Both your first sentence and that decomposition really express the same idea.

I have to go. bye

Pedro Tamaroff

@KarlKronenfeld Cheers. Thanks.

anon

03:22

I haz MO question.

@PedroTamaroff yes that is the geometry behind it

Pedro Tamaroff

@anon I am doing pretty well with this. Already halfway through the homework. =)

Actually 9/15 already done.

So a bit more.

@anon Reading.

Upvoted. =)

$\log_e$

WAI.

anon

to suggest the obvious question "what relation does $e$ have with $\Bbb Z$ that $q$ has with $\Bbb F_q[T]$?" to the reader

Pedro Tamaroff

@anon I realized as I read further.

anon

note that Cramer's heuristic does not predict PNT (rather, it is the other way around)

Pedro Tamaroff

"Cramer's heuristic"? Never heard of that.

anon

03:37

it is basically the heuristic that "n is prime with probability 1/log(n)" (n>1)

Pedro Tamaroff

@anon Ah.

anon

if you ask an asymptotic question about prime numbers then use Cramer's heuristic you should get the correct answers in simple situations. in fact it was something of a discovery when situations where it didn't work turned up.

Pedro Tamaroff

My respect for Landau is non-measurable.

U.U

@anon

anon

?

Pedro Tamaroff

@anon It have proven that if $A$ is Boolean, then $2x=0$ in $A$ and every prime ideal is maximal, with $A/\mathfrak p$ the field with two elements.

Now I have to prove that every fin gen ideal is principal. Is the proof complicated?

It is eluding me at the moment.

@FernandoMartin

Fernando Martin

03:49

I stopped at 9

I can think of it though

let's see

Pedro Tamaroff

@FernandoMartin 10 is trivial. =P

Fernando Martin

I haven't done 9 yet

Pedro Tamaroff

I did Ex.8 but forgot to do 11 in Ex.7.

@FernandoMartin It is not hard.

Fernando Martin

?

what do you mean by 11 in E7?

Pedro Tamaroff

@FernandoMartin Ex.7 asks to do 7 and 11 in AM.

I did 7 and forgot 11.

Fernando Martin

03:51

Oh, you mean Alicia's exercises

Pedro Tamaroff

Aha.

Fernando Martin

I don't know what edition she uses

I think the numbering isn't the same in every version - I'll check tomorrow

I was saying I solved up to 9 from A-M

Pedro Tamaroff

@FernandoMartin Oh.

I am doing all of them.

Fernando Martin

well let's see

Pedro Tamaroff

Don't care much about which edition she uses. =D

Fernando Martin

03:52

I know, but we'll need to know if we're meant to hand in the exercises

anyway

you get 2x=0 by looking at (x+x)^2

right?

anon

@PedroTamaroff not too much no

Pedro Tamaroff

@FernandoMartin $(x+1)^2$.

@anon OK

Fernando Martin

ahh, you're right

Pedro Tamaroff

I'll think harder.

anon

you want $\forall x,y\exists \pi:(x,y)=(\pi)$

Pedro Tamaroff

03:54

@anon Yeah, I noticed proving the case of two generators does it.

Fernando Martin

How did you solve 9?

the non-trivial implication

anon

hmm, I forgot why I wrote down $x\pi=x$ and $y\pi=y$

Pedro Tamaroff

@FernandoMartin We have an explicit formula for $\sqrt{ \mathfrak a}$.

Fernando Martin

I know, it's the intersection of all primes containing a

Pedro Tamaroff

So that does it.

Fernando Martin

03:55

That proves that if a is radical then it's an intersection of primes

Pedro Tamaroff

@FernandoMartin Yes.

Fernando Martin

Jesus I'm stupid

Pedro Tamaroff

But the other direction is trivial.

Chandan

Hi, can anyone help me with this problem? math.stackexchange.com/questions/744812/…

Fernando Martin

But that was proved on the text beforehand

Are we missing something?

Pedro Tamaroff

03:57

@FernandoMartin I don't understand what's the issue.

Fernando Martin

I'm just saying, it seems weird that the exercise asks you to prove something that is trivial

Pedro Tamaroff

Dunno.

Fernando Martin

Exercise 9 from Alicia's guide is really cool

Pedro Tamaroff

@FernandoMartin The proof is in AM. =P

Transcript for

$Mathematics$ Mathematics