Algebraic Number Field Puns

The solutions to certain quintics (or even higher order equations) with integer coefficients are not expressible using finite nested radicals. I believe these are called 'unsovlable' quintics.

But the solutions to these equations are still by definition algebraic numbers.

Edit: For people saying that numbers expressible in terms of finite terms of radicals should be called 'FOO' and thus what I want should be called non-FOO, non-FOO would also include the transcenyoudental numbers, something I explicitly don't want. So I guess you would have to call them non-FOO algebraic numbers.

Apparently, there is a correlation between Rieman's Hypothesis and Random Matrix theory, namely Montgomery's Pair Correlation. Evidently an analogue of Riemann's hypothesis, known as Weil's conjecture, which concerns generating functions over finite fields was proven by Deligne . Deligne's work does not involve random matrices.

Is there a "random matrix proof" of Weil's conjecture ? Is there a "random matrix proof" of Weil's conjecture for the simple case of the projective line ?

Is there an analogue of Montgomery's Pair Correlation for Weil's proven analogue of the Rieman Hypothesis ?

Note Weil's conjectured analogue of the Riemann Hypothesis is already proven. So, I am asking if we can work backwards and show over finite fields, there exist correlation among pairs of zeros for local Zeta's in the sense of Montgomery.

Not much more to the question. I know you can prove the existence of an algebraic closure using the axiom of choice. And for the rationals, you can construct it as a subfield of C the complex numbers. What can be done for the p-adic fields over Q?

I'm not sure about this one

I'm looking for some advanced undergraduate and graduate level algebraic number theory courses where the video lectures are available. Any recommendations are highly appreciated.

I need some help understanding basic algebraic number theory. Sorry if my questions seem stupid, I'm just kind of confused, right now.

For every integral domain there is an associated fraction field. This can be defined as the set of pairs (a,b) with an equivalence relation (a,b) ~ (c,d) iff ad = bc, right? Why is the condition of "integral domain" necessary? Does having zero divisors ruin things somehow?

An integral domain,R, is called "integrally closed" if any element in the associated field of fractions satisfies a polynomial in R[x]. For example, the set of all algebraic integers (solutions to polynomials in Z[x]) is integrally closed (proven in textbook).

So if A is the set of algebraic integers, and B is its fraction field, it follows that B = A, right? And that this is the case when A is any integrally closed integral domain?

Real parts of the roots*

I'd like to do an undergraduate course on algebraic number theory. I've already taken finite geometry and I'm planning on doing differential geometry. However, I'm in need of some background. What are good resources for that topic? I would love to read some of the papers as well. If I could read some literature on the subject as well, that would be even better.

(I am aware that number theory is not the best place to learn algebraic number theory, but I'm not sure about the other topics).

Thanks!

I know there is some overlap between both areas (eg L-functions, Dedekind zeta functions and the class number formula). But is there any reasearch on how algebraic NT tools can be used to solve analytic NT problems or viceversa? Or maybe some research area that uses both analytic and algebraic methods?

I am asking because I am about to apply for PhDs, and my interest is both algebraic and analytic NT. So I would love to get to know more about possible research directions.

I'm a high school student interested in learning algebraic number theory. My background is Group theory up to the sylow theorems, basic ring theory from Pinter's "a book of abstract algebra" and elementary number theory. I don't know if it's relevant but I know some analysis from the book called "yet another introduction to analysis" and also some basic linear algebra. I really like algebra and number theory (at least what I've studied till now) and I wish to learn algebraic number theory.

My question is what are the prerequisites for studying algebraic number theory (eg. how much field theory do I need to know) and what are some good resources/books for learning algebraic number theory.

Thanks in advance.

NOTE: I made a similar post couple of days ago on r/learnmath but it didn't got any attention so I'm posting here.

I think I figured it out myself :) My solution is in the comments.

I've managed to find a solver online to see that there are in fact no integer solutions, but I'm interested in proving why this is the case.

I've made some progress so I'll walk through it, and then offer some questions as to where I'm lost and/or confused.

x^(2) + 29y^(2) is the norm of x + y√-29 in the field K = Q( √-29 ), with ring of integers R = Z[ √-29]

(this is because -29 ≡ 3 mod 4)

If we consider the principal ideal in R, I = (x + y√-29), this ideal also has norm 2 * 3^(50) . Since R is a Dedekind domain, we can factor I into a unique product of prime ideals up to reordering. The ideal norm is multiplicative, so we first factor the ideals (2) and (3) into prime ideals in R, since these ideals have norms 2^(2) and 3^(2) respectively.

Using the Dedekind Prime Factorisation Theorem we can show that (2) = P^(2) where P = (2, 1 + √-29), and that (3) = QQ where Q = (3, 1 + √-29) (and where Q represents the conjugate ideal). The DPF Theorem also tells us that N(P) = 2, and N(Q) = N(Q) = 3.

Here, I then want to say that we conclude I = P * Q^(50 - t) * Q^(t) , and thus we have 51 options for t, and then another 2 options from the units in R. However I know this is wrong since the answer is 0, and not 102.

I'm pretty sure the issue is the fact that the class group of R is non-trivial and therefore for some (or all I suppose) value of t, we end up with a contradiction where we claim a principal ideal (I) is equal to some non-principal ideal, P * Q^(50 - t) * Q^(t). I know that the class number here is 6, since it's easy to search online. I'm really inexperienced with using the class group to solve problems, so I'm stuck on where to go from here. I've also noticed that if I change the equation to x^(2) + 29y^(2) = 2^(2) * 3^(50) (ie, add a factor of 2), then the problem has many solutions.

Ideally, I'd like a bit of a walkthrough for the remaining bit of the problem, and then have it highlighted where that extra factor of 2 actually changes our answer.

Let M = {

 0 0 
ia a

with the propriety that a is a complex number}

Show that (M, +, *) is a comutative field.

The first thing that drew my attention is the fact that I have to show that (M, *) is a group. This means that the identity element is part of M. This doesn't check out. I2 =

1 0
0 1

is not part of M since the top 2 numbers are 0 in all elements of M. Since I2 is the identity element when multiplying matrices then doesn't this mean that (M, +, *) is not a field and that (M, *) is only a semigroup?

For the sake of an example, we will consider the finite field [;\mathbb{F}_{17}\cong\mathbb{Z}/17\mathbb{Z};]. By elementary number theory theorems relating to quadratic residues, it is fairly simple to show that [;3;] is not a quadratic residue modulo [;17;]. Does it make sense to talk about the field [;\mathbb{F}_{17}(\sqrt{3})=\{a+b\sqrt{3}|a,b\in\mathbb{F}_{17},\sqrt{3}^{2}=3\};]? Is it useful for anything at all?

Edit: Getting LaTeX, MathJax and reddit formatting to all play nicely together is always fun at 1:30AM.

I've been using scalaz for that sweet |+| operator from Semigroup on multiple occasions. I'm surprised that I can't find actual Group (or Field for that matter) structure here.

I would like to use |-| (or negation). Is there an extension I can't find or another library that provides such algebraic structures?

So, let's assume a vector space V=R^n over the field K of real numbers, K=R. Now let's define an arbitrary linear operator t: R^n -> R^n with its matrix A, such that t(x) = Ax for any vector x in R^n.

The eigenvalues of a matrix are defined as a scalar value by which our corresponding eigenvector is scaled and since we're working over the field of real numbers, one would expect any eigenvalue to be real. However, as we all know, it's really simple to end up with a complex eigenvalue, even for some simple matrices. How is this possible?

What algebraic topics are required for keeping up with modern trends in analytic number theory?

Thanks in advance!

Thought this was a pretty interesting little stat, he’s shot 25% or lower in about 18% of his games this year. Could be due to shooting a higher % of threes, but still kinda unexpected.

Source

If we restrict ourselves to [; \mathbb{Q} ;] algebraic numbers are [; a \in \mathbb{C} ;] such that there is a polynomial [; f \in \mathbb{Q}[X] ;] such that [; f(a)=0 ;] , this forms the field of so called algebraic numbers. But what if we generalize this idea to power series? Maybe something like: [; a \in \mathbb{C} ;] is called " [; \infty ;]-algebraic" if there exists [; f \in \mathbb{Q}[[X]] ;] such that [; f(a)=0 ;] converges to zero in the [; \mathbb{C} ;] with the usual metric. This kind of definition poses some convergence questions and loses a whole lot of nice properties but it's still interesting. For example [; \pi ;] is [; \infty ;] -algebraic since [;\sin(\pi)=0 ;] . Overall you get some interesting questions: like do the [; \infty ;] -algebraic numbers of [; \mathbb{C} ;] form a field? (they seem to form a commutative ring) Are there complex numbers which are not [; \infty ;] -algebraic ?

Is there any work done on this or is this just a useless definition?

Edit: Thanks for your answers! It seems that these weird numbers turn out to be all the complex numbers

20 minutes

6 Points

3 assists

2 rebounds

2 turnovers

2-14 from the field

0-9 from three

Source

Hiya, I'm a math undergrad and I'm currently in love with algebra, and so I thought it would be neat to make a little post about how much I love algebraic numbers. I talked to one of my friends about how they are just roots of integer polynomials and I said for quadratic irrationals you pick 3 integers "and get a super cool new number for free!" Another really cool thing is that algebraic numbers are countable, and they form a field. Anyways, just showing the best numbers some love, ciao~