On the Ring of Integers in an Algebraic Number Field as a representation Module of Galois Group

Abstract

1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are: (I) identical representation,(II) rationally irreducible representation of degree l – 1,(III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

On the Galois Cohomology Group of the Ring of Integers in a Global Field and its Adele Ring

“…In the same way on the algebraic number field of Yokoi [2], the following lemma and proposition are proved also on algebraic function fields in one variable over a finite constant field and their local fields.…”

On a Characterization of the First Ramification Group as the Vertex of the Ring of Integers

“…To achieve our aim. we use the cohomological characterization of the tamely ramified extension which H. Yokoi gave in [6]. We state this characterization as Theorem 2 in this paper.…”

Normal Bases in Galois Extensions of Number Fields

“…Frohlich [3] using "Kummer invariants" considered the case when K\F is a Kummer extension and gave necessary and sufficient conditions that O^ have a normal basis. Yokoi [11] using the structure theory of integral representations of cyclic groups of prime order described the integral representations afforded by O^, KjQ a cyclic extension of prime degree. Chapter II gives necessary conditions for an ambiguous ideal in a wildly ramified Galois extension to have a normal basis; among them is the triviality of a second ramification group.…”