# Abstraction

#### Month: September 2016

The book “Number Fields” by D. Marcus is a very well-known introductory book on algebraic number theory. Its most memorable aspect is, without a doubt, the great number of exercises it contains. They vary from short(ish) computational exercises, through various technical results used later in the book, to series of exercises aimed to establish (sometimes very deep) results in number theory. They are structured in a way which allows even an unexperienced reader be able to solve most, if not all, exercises even on their first reading, thanks to (often very elaborate) hints provided.

However, even then a reader might want to refer some external source in order to see how the exercise can be solved, because otherwise it might be difficult to proceed any further (I myself would appreciate such a source at times). And, as they say, if you want something done right, do that yourself.

This post is based on Marcus’s Number Fields. More specifically, it is based on a series of exercises following chapter 4.

Recall the definition of the intertia group of a prime $$\frak P$$ in $$\mathcal O_L$$ lying over a prime $$\frak p$$ in $$\mathcal O_K$$ ($$L/K$$ is a Galois extension of number fields) – it’s the set of all $$\sigma\in G=\mathrm{Gal}(L/K)$$ such that, for all $$\alpha\in L$$, we have $$\sigma(\alpha)\equiv\alpha\pmod{\frak P}$$. We now generalize this group.

Definition: In setting as above, we define the $$n$$th ramification group $$E_n$$ to be the set of all $$\sigma\in G$$ such that $$\sigma(\alpha)\equiv\alpha\pmod{\frak P^{n+1}}$$. The groups $$E_n,n>1$$ are called the higher ramification groups.

Discriminant of a number field is arguably its most important numerical invariant. Quite closely connected to it is the different ideal. Here we discuss the most basic properties of these two concepts. The discussion mostly follows this expository paper by K. Conrad, but also takes from a series of exercises in chapter 3 of Number Fields.

This blog post consists of three parts. The first of them contains a somewhat nontechnical description of Riemann hypothesis. In the second one we discuss what the “correct” analogue of Riemann hypothesis is for polynomials over a finite field. Finally, in the last section, we prove the Riemann hypothesis for polynomials.

Alice: Fine, but I want to move first.
Bob: What? This is my game!
A: Precisely! I’m sure you have figured out the strategy to win by now, so let me at least enjoy the game for a little bit.
B: Alright, fine. Are you sure you’ve got all the rules?
A: It’s not like there are too many of them.
B: So we first agree on the number of moves $$q$$ and another number $$p$$.
A: Sure enough. Then I choose a number $$x_1$$ and take its square…
B: …then I choose another number $$x_2$$ and add its square to yours…
A: …and we just keep adding squares, summing $$q$$ of them in total.
B: That’s right, and you want to prevent the total sum from being a multiple of $$p$$.
A: It doesn’t sound like a very exciting game, I don’t really think I want to play it.
B: Really? Not even once?
A: I mean, that’s just adding numbers! And after all, shouldn’t we be sending encrypted messages to each other or something instead of playing games?
B: You know, cryptography can be a bit like a game as well… either way, I just wanted to do something else for one. What should I do with the game now? Just forget about it?
A: If you care about your game so much we can try to work something out with it without playing it.

The Dirichlet L-functions are an extremely important tool in studying primes in arithmetic progressions – their zeros “control” distribution of primes in arithmetic progressions in the same manner as zeros of Riemann zeta function control the overall distribution of primes. The first and the most elementary result involving these zeros, which is the key result in most proofs of Dirichlet’s theorem, is that there is never a zero at point $$s=1$$. This post will present a proof of this fact using results from algebraic number theory.

This post is based solely on the content of Marcus’s Number Fields. The prerequisities for it are basic results about ideals in number fields and a minute amount of complex analysis. No background in analytic number theory is necessary.

We shall establish relations between degrees, inertia degrees and ramification indices involved in decomposition and inertia field of a given prime in Galois extension. This is based on Marcus’s Number Fields and online notes by R. Ash. It follows a very similar appoach to, but is not based on, the one which can be found in this blog post by Sander Mack-Crane. For this blog post, understanding of Galois theory and basic facts about number fields is necessary.

Based on arguments in Marcus’s Number Fields and K. Conrad’s expository paper on ideal factorization. I assume familiarity with the concepts related to ideals and fractional ideals in a commutative ring.

Recall that an integral domain $$R$$ with the field of fractions $$K$$ is called a Dedekind domain if the following conditions hold:

• $$R$$ is Noetherian, so every nonempty set of ideals has a maximal element, or equivalently, every ideal is finitely generated,
• every prime ideal in $$R$$ is a maximal ideal, and
• $$R$$ is integrally closed in $$K$$, so that every root of a monic polynomial from $$R[x]$$ lying in $$K$$ lies in $$R$$.

Our goal is to prove the following proposition:

Proposition: Assume $$R$$ is a Dedekind domain. For any nonzero (i.e. containing a nonzero element) fractional ideal $$I$$ in $$K$$ there exists a fractional ideal $$J$$ such that $$IJ=R$$.