Basic Concepts of Algebraic Number Fields

Abstract

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).