evening

Hello guys. Can you recommend me the best supplement to Pinter, please.

@Everstudent what about it are you wanting to supplement? What does Pinter lack that you want/need?

@Astyx maybe something like this is more feasible for what I just described: $$\forall \alpha \in \mathbb{N},\forall \omega \in \mathbb{N}, \exp(x)=\sum_{\alpha}^{\infty}{\delta_{\alpha}(x)},(\exists g(x)=\prod_{\omega}^{\alpha}\delta_{\omega}(x) \text{ for some }\alpha : \exp(x) = g(x))$$.

Because then, exp(x) is reduced to a finite product that may or may not be tethered to x, and exp is of course a repeated multiplication of e, but for all real values of x rather than just integers.

-Dummit&Foote

@Everstudent kconrad.math.uconn.edu/blurbs

Look at the group theory section.

Other than that - I guess, more motivation as to why and what is important. For example - before checking out other materials, I did not get that Group Actions are important at all or Conjugacy - I mean just 5 exercies at the end of the chapter in the middle of other exercises which are not that significant

Especially "Group Actions" and "Transitive Group Actions".

Conrad is a great pedagogue.

Then the question is whether or not such a form can be used to compute $e^{ix}$ and from there, the real and imaginary parts individually as real-valued functions of x.

So to ask a more concrete question - what do I need to properly understand/prove Sylow - besides Group Actions/Homomorphisms/Normal groups/Conjugacy?

Thanks @anakhro, I will check it out :)

He also has a proof of Sylow's theorems there and more on them in another one.

Really good at writing, I can't recommend him enough.

yeah, Conrad's stuff is great

But also a good thing to grok Sylow's theorems is by doing exercises on it. So try Dummit & Foote for exercises (google might have a scanned copy lying around).

It's kind of an expensive book, though.

But you can also google "sylow theorem exercises pdf" or something like that and see if anything good pops up.

And just sit there and struggle with seeing how to use Sylow's theorems.

@Balarka @Alessandro Death Atlas - Cattle Decapitation

really good album

death, atlas, cattle, and decapitation are not words I expect to hear in one sentence

Dummit & Foote is good but for some reason it has a bad rep

but I liked it

Are the Sylow theorems useful for say understanding Polynomial rings/Galois groups? I am trying to get enough knowledge to do a bit of Algebraic Number Theory

uh oh

They're good for determining Galois groups

Dummit & foote is good for the number of exercises as well as being a good reference for stuff you ought to know by grad school. But pedagogically I think it is remarkably weak. But then again, so are most algebra books.

So are most math books. :((((((((

@Everstudent good choice of subject btw

I am not sure where to begin, I mean I did a lot of Elementary Number Theory, but I got stuck on Quadratic Reciprocity, that was like ages ago. So I did not like the elementary proofs of it - they seemed made up and random for my taste. So I started checking out the more Algebraic Proofs and found I do not understand them at all. So then I found out that besides commutative rings(I had some exposure to those) - there are groups, subgroups, morphisms,etc. So all that got me into Abstract Algebra.

There's a cool proof by cyclotomic fields which is a cool thing to aim for

So I tweaked my approximation today... the absolute difference of cos(x) and my cosine approximation is now recognizable as a function... all I did was make cos(1) the coefficient of $\sqrt{\frac{\pi^{2}}{4}-\left(\frac{x\pi}{2}\right)^{2}}$. No idea what, though. desmos.com/calculator/oamy1zn1rk

altho I don't actually remember the proof, just that it's really cool

So one of my issues is that I do not get the relation between topics - say for example I have an intuition that groups actions will be useful all the time even in rings. But for Sylow theorems - I have no idea what can I use those for.

So yeah, I guess a big problem with Pinter is the lack mention of the importance/usefulness of the presented material.

(Tweak the value of b to see how the difference [the purple plot] changes with the approximation [blue plot])

Sylow theorems are used in pretty much any structure result about groups

What about Algebraic Number Theory?

Also in algebraic number theory you meet the sentence "consider the $p$-part of [insert interesting invariant]" often

Hm, so is there a book which gives a more of an overview of things. Kinda like a reference, a short one, without proofs - just explaining things and giving links between them

erm

There's Ian Stewart's Algebraic Number Theory which is a more elementary introduction but is still a textbook

So I can get some structure in my head before trying to add information. Kinda like preparing boxes with names on them I guess.

You can also look at Harold Edwards' "A Genetic Introduction to Algebraic Number Theory" for the development of the subject, and there's Alaca & Williams Introductory Algebraic Number Theory which is also quite elementary

Hm, by elementary, you mean it's not that deep or that it's not that algebraic/abstract?

I mean, it's written as a "first treatment" of the theory, so it doesn't have lots of the technicalities

I think in that book he develops just enough theory to prove Fermat's last theorem for regular primes

in one of the cases

(By that book, I mean Ian Stewart's book)

@Everstudent also encyclopediaofmath.org/wiki/Algebraic_number_theory

is a cool article

I missclicked ignore on a user

How do I unignore

hope it wasn't me cuz nobody else likes number theory here and I'm latching onto you in the hope that you'll join chat and I can have a friend

heh there is Gowers's Princeton Companion.

I found the un-ignore button :D

1000 pages to summarize your undergrad.

@EdwardEvans I guess I have a lot to learn in Abstract Algebra, before I can dive into Algebraic Number Theory, thanks for all the references/links

as long as you know what rings, groups, fields, and Galois groups are, you can read Ian Stewart's book

@Edward hey I'll do ANT next time they actually offer it here

good man

I know basic things about rings, groups, fields. But I do not know about Galois Groups yet.

@Thorgott what do you mean by offer it, like if someone here asks a question or

There was a dynamical proof of the prime number theorem that I had seen

It was ultra cute but now I can't find the book

no, I mean I'll attend an ANT lecture at my uni the next time one is offered

hm, I take back what I said earlier. I wasn't paying attention and forgot to multiply x in cos(x) by pi/2.

But, I did find that I have significantly more room to play with concerning the error function now that I have a coefficient of cos(1)sin(1) now.

hey chat.

I have to explain to 12yo students that the product of 2001 (in fact, any odd number of) transpositions is never the identity, but I can't talk about groups or the discriminant $\Delta = \prod\limits_{i < j} x_i - x_j$.

How would you guys do it? I'm not finding any combinatorial argument.

Formally it's not that hard to show that the function $sgn(f) = (-1)^{inv(f)}$ is a "group homomorphism" (you don't have to use these words, but that's the underlying reason).

And so then you just show that sgn(id) = 1 and sgn(transposition) = -1.

But that doesn't actually give you too much intuition to play off of, especially for 12 year olds.

But complimented the law of small numbers (i.e. "what if I do 1 transposition? 2? 3? 4? ...?")

maybe it might be a good backup plan.

@LucasHenrique had you considered that method before?

yup, but I was looking for a proof. :p

well, don't you basically need to show them that the identity transposition is always even, I mean why pick 2001

you can do that by showing them that you can take some random transposition expression of the identity - say 2k+1 ones, and you can always remove two transpositions. so if it's odd

@LucasHenrique Do you know of the proof that I referred to above with the sgn function?

you will be left with one transposition at the end

but the identity cannot be equal to (ab) so there you go

meaning you take $(a_{1}a_{2})...(a_{k+1}a_{k+2) a total of 2k+1 transpositions, and you show that you can always decrease their number by two

so if we start with an odd number we get 1 transposition at the end but the identity cannot be equal to a single transposition

It's a popular proof that doesn't require groups/homomorphisms/orbits/etc.

@LucasHenrique why should 12 yr olds care?

we're having preparatory olympiad questions

@Everstudent I think you have to technically go over something about "homomorphisms" in essence.

wait, it was actually reducing them by one. I guess my memory on this was muddy. So the idea was that you're looking for the last occurence of x in the transpositions. And everytime you bring x closer to the first transposition, removing other x's in the process. So in the end we get the first transposition to contain x while no other transpositions contain it. So obv it can't be the identity.

I have no idea what olympiad for 12 yr olds is.

one of them is: "given the list of numbers 1, 2, ..., n, you can interchange the position of any two of them. can you get the same list after 2001 permutations?"

mathonline.wikidot.com/the-identity-permutation

This is the proof I was describing, it's popular.

Maybe a contradiction pf with well-ordering on odds?

That's the proof I have in my algebra book, in fact, I think.

Determinant is a cheating proof, because well-definedness of det is deep.

That's what everstudent is referring to, but it's a bit sneaky.

Am I doing something wrong? I'm trying to solve for the exact value to use for b for my approximation to determine the max value such that as x approaches $\pm 1$ from 0, the function evaluates to zero at $\pm 1$. I don't know how to do that. I end up with $b = e^{0} = 1$/, but ln(1) is zero which would make everything zero but in the wrong way. desmos.com/calculator/vzfu2bofxg

Or rather that $f(x)$ should evaluate to zero when $x = \pm 1$ unless $\pm 1$ are the limits of $f(x)$, but I wouldn't know how to evaluate the limits of $f(x)$ using $\pm 1$.

@anakhro maybe it's in Furstenberg's book?

He surely has dynamical proofs of Ramsey type results in it

Seriously, how do I compute minima and maxima using a limit? I mean look at this mess here; I run out of computation time: wolframalpha.com/input/…

No, wait, this one actually shows a graph and it looks like the beginning of a bad math-themed horror movie: wolframalpha.com/input/…

I told myself I wouldn't work on this sine and cosine stuff anymore after today (unless something comes up) until I have time for it in recreation because I need to get other things accomplished, so it's now or... significantly later.

Anyone?

Hey guys, did you know that a real-valued function of x takes True as a valid number?? wolframalpha.com/input/…

@AlessandroCodenotti sounds sort of familiar I will have a look later.

are there any analysts in the crowd?