Algebraic Number Theory

Introduction

At its core, the ancient subject of number theory is concerned with the arithmetic of the integers. The Fundamental Theorem of Arithmetic, which states that every positive integer factors uniquely into a product of prime numbers, was contained in Euclid’s Elements, as was the infinitude of the set of prime numbers. Over the centuries, number theory grew immensely as a subject, and techniques were developed for approaching number-theoretic problems of a various natures. For instance, unique factorization may be viewed as a ring-theoretic property of $\mathbb{Z}$, while Euler used analysis in his own proof that the set of primes is infinite, exhibiting the divergence of the infinite sum of the reciprocals of all primes.

Algebraic number theory distinguishes itself within number theory by its use of techniques from abstract algebra to approach problems of a number-theoretic nature. It is also often considered, for this reason, as a subfield of algebra. The overriding concern of algebraic number theory is the study of the finite field extensions of $\mathbb{Q}$, which are known as number fields, and their rings of integers, analogous to $\mathbb{Z}$.

The ring of integers $𝒪$ of a number field $F$ is the subring of $F$ consisting of all roots of all monic polynomials in $\mathbb{Z}[x]$. Unlike $\mathbb{Z}$, not all integer rings are UFDs, as one sees for instance by considering the factorization of $6$ in the ring $\mathbb{Z}[\sqrt{-5}]$. However, they are what are known as Dedekind domains, which have the particularly nice property that every nonzero ideal factors uniquely as a product of nonzero prime ideals, which are all in fact maximal. In essence, prime ideals play the role in $𝒪$ that prime numbers do in $\mathbb{Z}$.

A Dedekind domain is a UFD if and only if it is a PID. The class group of a Dedekind domain is roughly the quotient of its set of nonzero ideals by its nonzero principal ideals, and it thereby serves as something of a measure of how far a Dedekind domain is from being a principal ideal domain. The class group of a number field is finite, and the classical proof of this is in fact a bit of analysis. This should not be viewed as an anomalous encroachment: algebraic number theory draws heavily from the areas it needs to tackle the problems it considers, and analysis and geometry play important roles in the modern theory.

Given a prime ideal $𝔭$ in the integer ring $𝒪$ of a number field $F$, one can define a metric on $F$ that measures the highest power of $𝔭$ dividing the difference of two points in $F$. If the finite field $𝒪∕𝔭$ has characteristic $p$, then the completion $F_{𝔭}$ of $F$ with respect to this metric is known as a $p$-adic field, and the subring ${𝒪}_{𝔭}$ that is the completion of $𝒪$ is called its valuation ring. In the case that $F=\mathbb{Q}$, one obtains the $p$-adic numbers ${\mathbb{Q}}_p$ and $p$-adic integers ${\mathbb{Z}}_p$. The archimedean fields $ℝ$ and $ℂ$ are also completions of number fields with respect to the more familiar Euclidean metrics, and are in that sense similar to $p$-adic fields, but the geometry of $p$-adic fields is entirely different. For instance, a sequence of integers converges to $0$ in ${\mathbb{Z}}_p$ if and only it is eventually congruent to zero modulo arbitrarily high powers of $p$.

It is often easier to work with $p$-adic fields, as solutions to polynomial equations can be found in them by successive approximation modulo increasing powers of a prime ideal. The “Hasse principle” asserts that the existence of a solution to polynomial equations in a number field should be equivalent to the existence of a solution in every completion of it. (The Hasse principle does not actually hold in such generality, which partially explains the terminology.)

Much of the formalism in the theory of number fields carries over to a class of fields of finite characteristic, known as function fields. The function fields we consider are the finite extensions of the fields of rational functions ${𝔽}_p(t)$ in a single indeterminate $t$, for some prime $p$. Their “rings of integers”, such as ${𝔽}_p[t]$ in the case of ${𝔽}_p(t)$, are again Dedekind domains. Since function fields play a central role in algebraic geometry, the ties here with geometry are much closer, and often help to provide intuition in the number field case. For instance, instead of the class group, one usually considers the related Picard group of divisors of degree $0$ modulo principal divisors. The completions of function fields are fields of Laurent series over finite fields. We use the term “global field” refer to number fields and function fields in general, while the term “local field” refers to their nonarchimedean completions.

An introductory course in algebraic number theory can only hope to touch on a minute but essential fraction of the theory as it is today. Much more of this beautiful edifice can be seen in some of the great accomplishments in the number theory of recent decades. Chief among them, of course, is the proof of Fermat’s last theorem, the statement of which is surely familiar to you. Wiles’ proof of FLT is actually rather round-about. It proceeds first by showing that a certain rational elliptic curve that can be constructed out of a solution to Fermat’s equation is not modular, and then that all (or really, enough) rational elliptic curves are modular. In this latter aspect of the proof are contained advanced methods in the theory of Galois representations, modular forms, abelian varieties, deformation theory, Iwasawa theory, and commutative ring theory, none of which we will be able to discuss.