An Alternative Proof of a Generalized Principal Ideal Theorem

Abstract

Recentry Mr. Terada1> has proved the following generalized principal theorem : Theorem. Let K be the absolute class field over k, and Q a. cycic intermediate field of K/k, then all the ambigous ideal classes of Q will become principal in K. I also generalized this theorem to the case of ray class field.2> By using Artin's law of reciprocity we can state above theorem in terms of the Galois group, and we have Theorem. Let G be a metabelian group with commutator subgroup G', H be an invariant subgroup of G with the cyclic quotient group G/H, and A element of H with ASA -ls-1eH' (S being a generator of G/H), then the "Verlagerung" V(A) = IITATAfrom H to G' is the unit element of G. Thereby T runs over a fixed representative system of G/ H, and T A means the representative corresponding to the coset TAG'. At first we tried to solve this by means of Iyanaga's method depending upon Artin's splitting group,3> which is generated by G' and the symbols Aa(A1 = 1, u~r = G/G'), and with r as operator system by rules (1) (2) U" = SaUS; 1 (U€G'), A~ = A;1A"'"D;,~ , S" being the representative of GIG' corresponding to usr, and . (3)