On the deduction of the class field theory from the general reciprocity of power residues

Abstract

We denote by (A) Artin's reciprocity law for a general abelian extension of a finite degree over an algebraic number field of a finite degree, and denote two special cases of (A) as follows: by (AC) the assertion (A) where $K/F$ is a cyclotomic extension; by (AK) the assertion (A) where $K/F$ is a Kummer extension. We will show that (A) is derived from (AC) and (AK) only by routine, elementarily algebraic arguments provided that $n = (K : F)$ is odd. If $n$ is even, then some more advanced tools like Proposition 2 are necessary. This proposition is a consequence of Hasse's norm theorem for a quadratic extension of an algebraic number field, but weaker than the latter.