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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).

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra’s construction of independent cyclotomic units.

Letpbe a prime. If one adjoins toQallpn-th roots of unity forn= 1,2,3, …, then the resulting field will contain a unique subfieldQ∞such thatQ∞is a Galois extension ofQwith Gal (Q∞/Q)Zp, the additive group ofp-adic integers. We will denote Gal (Q∞/Q) byΓ. In a previous paper [6], we discussed a conjecture relatingp-adicL-functions to certain arithmetically defined representation spaces forΓ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces forΓ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

In this paper, we shall give explicitZ-basis of certain maximal orders of definite quaternion algebras over the rational number fieldQ(See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. LetDbe a division quaternion algebra overQ.

Let k be a p-adic number field and o be the ring of all integers of k. Let K/k be a cyclic ramified extension of prime degree p with Galois group G. Then the ring of all integers of K is o[G]-module. The purpose of this paper is to give a necessary and sufficient condition for o[G]-modale to be indecomposable.