Hello everyone! I am sorry to inform you, but this is the last post I am going to make on this blog... because I have moved to a different site! From now on, my blog will be hosted on my university server, and you can find it here. If you are interested in the reason …

# Proof sketch of Thue’s theorem

In this post a proof of the following theorem is going to be sketched, following the treatment in Borevich and Shafarevich's Number Theory. This sketch is by no means meant to be highly detailed and I am writing it mostly for my own purposes, so I avoid proving some things, even if they aren't that straightforward. Thue's …

# Abstract divisors

The goal of this blog post is to provide an overview of the general theory of divisors, as described in a book Number Theory by Borevich and Shafarevich. The point of this post is for it to be somewhat expository, so it will avoid the longer proofs, sometimes just sketching the ideas. Throughout, by a "ring" …

# IP=PSPACE

Complexity theory is a mathematical study of algorithmic problems with regard to how cost-effective solutions they have. Most prominent are the decision problems, which are yes-no questions for which answer depends on some further input, for example, "Is $latex n$ a prime number?" is a decision problem depending on the input $latex n$. I assume that …

# Starting a new project – Solutions to exercises in Marcus’ book (+some info)

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 …

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# Number of irreducible polynomials over a finite field

Warning to PROMYS students: this blog post contains major spoilers regarding problems 4 and 6 of the Open Door Problem Set. Read at your own risk. The following problem has appeared as problem P3 on the short exam #2 during PROMYS Europe 2016: Let $latex p,n\in\mathbb N$ with $latex p$ prime. How many irreducible polynomials …

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# Higher ramification groups

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 $latex \frak P$ in $latex \mathcal O_L$ lying over a prime $latex \frak p$ in $latex \mathcal O_K$ ($latex L/K$ is a Galois extension of …

# Discriminant and different

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. Discriminant …

# Proof of the Riemann hypothesis… for polynomials

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. Riemann zeta function and …

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# Quadratic reciprocity: the game

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 …