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.
Citation
Tomio Kubota. Satomi Oka. "On the deduction of the class field theory from the general reciprocity of power residues." Nagoya Math. J. 160 135 - 142, 2000.
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