The Zeta Function of an Algebraic Number Field and Some Applications

Abstract

At the end of Sect. 4.6 of Chap. 4, we left ourselves with the problem of determining the finite nonempty subsets S of the positive integers such that for infinitely many primes p, S is a set of non-residues of p. We observed there that if S has this property then the product of all the elements in every subset of S of odd cardinality is never a square. The object of this chapter is to prove the converse of this statement, i.e., we wish to prove Theorem 4.12. The proof of Theorem 4.12 that we present uses ideas that are closely related to the ones that Dirichlet used in his proof of Theorem 4.5, together with some technical improvements due to Hilbert. The key tool that we need is an analytic function attached to algebraic number fields, called the zeta function of the field. The definition of this function requires a significant amount of mathematical technology from the theory of algebraic numbers, and so in Sect. 5.1 we begin with a discussion of the results from algebraic number theory that will be required, with Dedekind’s Ideal Distribution Theorem as the final goal of this section. The zeta function of an algebraic number field is defined and studied in Sect. 5.2; in particular, the Euler-Dedekind product formula for the zeta function is derived here. In Sect. 5.3 a product formula for the zeta function of a quadratic number field that will be required in the proof of Theorem 4.12 is derived from the Euler-Dedekind product formula. The proof of Theorem 4.12, the principal object of this chapter, is carried out in Sect. 5.4 and some results which are closely related to that theorem are also established there. In the interest of completeness, we prove in Sect. 5.5 the Fundamental Theorem of Ideal Theory, Theorem 3.16 of Chap. 3, since it is used in an essential way in the derivation of the Euler-Dedekind product formula.