Vertices of ideals of a -adic number field II

Abstract

Letkbe a-adic number field with the ring 0 of all integers ink, andKbe a finite normal extension with Galois groupG.∏denotes a prime element of the ringof all integers inK. Then, an ideal (∏a) ofis an 0G-module. E. Noether showed that ifK/kis tamely ramified,is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows:is relatively projective with respect to a subgroupSofGif and only ifS⊇G1whereG1is the first ramification group ofK/k. Now we define the vertexV(∏a) of (∏a) as the minimal normal subgroupSofGsuch that (∏a) is relatively projective with respect to a subgroupSofG(cf. § 1). Then, the above generalization by A. Fröhlich impliesV() =G1. In the previous paper, we provedG1⊇ V(∏a) ⊇ G2, (whereG2is the second ramification group ofK/k(cf. Theorem 5). Further, we dealt with the case whereG=G1is of orderp2, and proved that ifV(∏a) ≠G1thena≡ 1(p2) andt2≡ 1(p2) for the second ramification numbert2ofK/k(cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degreepn(Theorem 7).