6 CONTENTS Preface This book is based on notes I created for a one-semester undergraduate course on Algebraic Number Theory, which I taught at Harvard during Spring 2004 and Spring 2005. The textbook for the first course was chapter 1 of Swinnerton-Dyer's book [SD01]. The first draft of this book followed [SD01] closely, but the current version but adding substantial text and examples to make t...

1 Number Fields 2 1.1 Norm, Trace, and Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Dedekind Rings 7 2.1 Fractional Ideals and Unique Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 The Ideal Class Group ...

Abstract: Inspired by the ”rough classification” ideas from additive combinatorics, Soundararajan and I have recently introduced the notion of pretentiousness into analytic number theory. Besides giving a more accessible description of the ideas behind the proofs of several wellknown difficult results of analytic number theory, it has allowed us to strengthen several results, like the PolyaVino...

Noncommutative geometry is a modern field of mathematics begun by Alain Connes in the early 1980s. It provides powerful tools to treat spaces that are essentially of a quantum nature. Unlike the case of ordinary spaces, their algebraof coordinates is noncommutative, reflecting phenomena like the Heisenberg uncertainty principle in quantum mechanics. What is especially interesting is the fact th...

Students having had a semester course in abstract algebra are exposed to the elegant way in which finite group theory leads to proofs of familiar facts in elementary number theory. In this note we offer two examples of such group theoretical proofs using the action of a group on a set. The first is Fermat’s little theorem and the second concerns a well known identity involving the famous Euler ...

This note contains some disconnected minor remarks on number theory . 1 . Let (1) Iz j I=1, 1<j<co be an infinite sequence of numbers on the unit circle . Put n s(k, n) _ z~, Ak = Jim sup I s(k, n) j=1 k=oo and denote by B k the upper bound of the numbers I s(k,n)j . If z j = e 2nij' a =A 0 then all the Ak 's are finite and if the continued fraction development of a has bounded denominators the...

The previous speaker concluded his address with a reference to Dedekind and Weber. It is therefore fitting that I should begin with a homage to Kronecker. There appears to have been a certain feeling of rivalry, both scientific and personal, between Dedekind and Kronecker during their life-time; this developed into a feud between their followers, which was carried on until the partisans of Dede...

10. Dirichlet characters and Dirichlet L-functions Definition 10.1. Let m ∈ N. A Dirichlet character modulo m is a function χ : N→ C that satisfies the following three conditions. (1) χ is periodic with period m, i.e. if a ≡ b (mod m) then χ(a) = χ(b). (2) χ is completely multiplicative, i.e. χ(ab) = χ(a)χ(b) for all a, b ∈ N. (3) χ(a) = 0 if and only if gcd(a,m) > 1. For m ∈ N we let (Z/mZ)× d...

The aim of this work is to investigate some arithmetical properties of real numbers, for example by considering sequences of the type ([bα]) , n = 1, 2, . . . where b ∈ N, α ∈ R, the terms of the sequences being in arithmetical progression, square-free, sums of two squares or primes. The results are most commonly proved for almost all α ∈ R or (α1, . . . , αm) ∈ R m (in the sense of Lebesgue me...