An Introduction to Algebraic Number Theory

This chapter consists of elementary number theory and deals with the greatest common divisor, the euclidean algorithm, congruences, linear equations, primitive roots, and the quadratic reciprocity law. The material covered here corresponds to the first four chapters of Gauss’s Disquisitiones arithmeticae (1801) and to the whole volume (70 pages) of Weil’s Number Theory for Beginners (1985). Equally small (95 pages) Bakers A Concise Introduction to the Theory of Numbers (1984) contains, in addition to those standards, quadratic forms, diophantine approximation, Fermat primes, Mersenne primes, Goldbach’s conjecture, twin primes, perfect numbers, the Riemann hypothesis, Euler’s constant, ζ(2n + 1), Fermat’s conjecture, Catalan’s conjecture, and so on. Following Weil, we regard this part of number theory as a rich source of structures of algebra such as groups, rings, and fields. Like computers in these days, the algebraic language is useful and economizes our thoughts. For example, group theory teaches us that the famous Fermat-Euler theorem on congruences is a special case of a simple theorem on finite groups and that the important existence theorem of primitive roots due to Gauss follows from the theorem stating that a finite subgroup of the multiplicative group of any field is cyclic. By doing so, number theory (elementary or advanced) is well ventilated. As for the proof of the reciprocity law, we followed the method of Gauss using Gauss sums because it is not only beautiful but also has a crucial influence on the later development of algebraic number theory.

In Chapter 2, we shall extend the results of Chapter 1 (number theory in ℚ) to the case of an algebraic number field K. Such a field K is a subfield of the field ℚ (the algebraic closure of ℚ) which is of finite dimension over ℚ. In particular, as a generalization of prime numbers in ℚ, we shall introduce the concept of prime ideals in K and extend Theorem 1.6 (fundamental theorem of number theory) to the case of number fields. This is the ideal theory of Dedekind (1831–1916). Next, we shall introduce the theory of Hilbert (1862–1943) which combines the theory of Galois (1811–1832) on field extensions with the ideal theory of Dedekind. We shall give an alternative proof of the Gauss reciprocity law (Theorem 1.27) by considering, for an odd prime I, the quadratic subfield ℚ\((\sqrt {{l^*}} )\), l* = (−1)^(l−1)/2l, of the lth cycloto-mic field K = ℚ\(({e^{2\pi ill}})\) (Remark 2.20). This situation will turn out to be a special case of the reciprocity in the theory of relative abelian extensions, i.e., the class field theory, due to Takagi (1875–1960) and Artin (1898–1962).

In Chapter 2, we used Dirichlet’s theorem on arithmetical pro-gressions in order to determine the kernel Ker _{k /ℚ} α of the Artin map of a quadratic field k/ℚ (Theorem 2.21) and promised the reader its proof in Chapter 3. Therefore this is at least one raison d’être of this chapter in the book. To prove Dirichlet’s theorem he invented the L-functions. The use of Dirichlet L-functions, however, goes beyond the proof of the theorem on arithmetical progressions. It turns out that the L-functions are closely related to the law of decomposition of rational primes in algebraic number fields. The reader will learn in Chapter 4 how to use L-functions to handle class numbers of some cyclotomic and quadratic fields. For this reason, we will not rush to a proof of the theorem of arithmetical progressions. We will rather start with basic topics such as geometry of numbers due to Minkowski (1864–1909) and the famous Dirichlet’s unit theorem.

The main purpose of this chapter is to derive the class number formula for subfields of the lth cyclotomic field [formulas (4.21) and (4.22)] by making use of analytical results in Chapter 3. In particular, for the quadratic subfield ℚ\((\sqrt {{l^*}} )\) of the lth cyclotomic field, one reaches the final formula due to Dirichlet [formulas (4.30) and (4.33)] using the famous determination of the Gauss sums (Theorem 4.2).