On the general principal ideal theorem
Katsuya Miyake · Proceedings of the Japan Academy Series A Mathematical Sciences · 1980
By means of the theory of the module of genus, S. Iyanaga and Herbrand established the general principal ideal theorem in [4], [5] and [6].In this paper, we prove the theorem in an improved form by an investigation of the structure of the idele groups [8].1. Let k be an algebraic number field, m a divisor of k, which may contain Archimedian primes, and K the ray class field modulo m of k.Denote the conductor, the different and the module of genus of K over k by [/, / and / respectively.Then as divisors of K, we have /=/./.For a prime ideal of K, let e()=e(/) be the order of ramification o over k, i.e. *()[(k).O, and *()+(k).O, and put K/ " k" e()-l= K/" (" )" e()-1.Our improved orm of the general principal ideal theorem is