The Galois‐equivariant Structure of the Norm Residue Symbol in Tamely Ramified‐extensions of 2‐adic Number Fields

Abstract

Let L be a p-adic number field of degree d over Q,,, which contains the p' roots of unity pps (s 2 1). Then V, = L x / ( L x ) p s is a free Z/p"Z-module of rank d + 2, and the @-norm residue symbol on L" induces on K, a hi-multiplicative form It is well known (see, e.g., [6]) that (-, -), is nondegeneratc and antisymmetric; if p = 2 and L doesn't contain a fourth root of unity, this form is non-alternating: there is an element a E L such that (a, a), = (a, -1), = -1. When L is a Galois extension of the subfield k with Galois group G, then (-, -), is Galois-equivalent: for a, h E 5, and g E G , (aY, bq), = (a, b): .Of special interest to us is the case where k is fixed and the Galois extension Llk is a tamely ramified of split type (defined in the next section). Here the Galois-module structure of VL has been explicitly determined, and precise knowledge of the symplectic structure of the norm residue symbol on this space when ps + 2 has led to a purely group-theoretical characterization of the absolute Galois group of k (KOCH [9]) and even to explicit presentations of this group (JAKOVLEV [3] and JANNSEN and WINCRERG [4] for p = I 2; ZELVENSKII [ 121 and DIEKERT [I] for p = 2 and s 2 2). This paper will address the remaining case, when k ( W ) / k is a ramified extension of the 2-adic number field k. In [8], H. KOCH obtained a structure Theorem for VL when L/k is unramified of 2-power order, and in [11] the author extended KOCH'S result to the case where L / k is tamely ramified abelian. The main result of this paper gives a characterization of the symplectic structure of V, for L/k tamely ramified of split type.