Local Class Field Theory: Perfect Residue Field Case

Abstract

The superconducting transition temperature T, has been determined for a number of technetium-based HCP solid solution alloys. Additions of V. Nb and MO raise Tc, whereas CO, Ni, Ru, and, in particular, Cr and Fe lower T, relative to pure technetium. The general trend of T, with the electron per atom ratio is consistent with that expected for a simple dependence on a rigid-band density of states, with the exception of that for the Cr and Fe alloys. An analysis of the data for these latter alloys in the context of the theory due to Kaiser would suggest that the enhanced depression of T, is due to the formation of a nonmagnetic resonance state. The hydrostatic pressure dependence of T, has been investigated up to 25 kbar with particular attention given to the HCP region of the Mo-Tc-Ru system. The observed variation of ?T,jdP is discussed in relation to current theoretical models.

Some remarks on the local class field theory of Serre and Hazewinkel

“…First we have K × /N L × = U K /N U L since L/K is totally ramified. We have short exact sequences This proposition shares large part with Fesenko's result in his paper [Fes93].…”

“…The authors would like to thank Professor Kato and Professor Taguchi for having helpful discussions, and to Professor Fesenko for suggesting relation between his work [Fes93] and our work. They also thank to the referee and Professor Suwa for reading a draft of the paper and giving comments for it.…”

Abelian localp-class field theory

“…Let .~,U be a normic subgroup in VK~n~Then N~Fl(~4 r) is a normic subgroup in VK~~ Proof It suffices to verify the assertion for a cyclic totally ramified extension L/F of degree p. Then the first and second properties of N~(JV) can be established similarly with the proof of Lemma 5 in[31] employing Lemma (2.6) of[7]. We will show that the class of normic subgroups coincides with the class of norm groups NL/F VKt,,~ of totally ramified galois extensions L/F,L C Fp.…”

“…2 of [7]). Recall that an additive polynomial over k is called k-decomposable if all its roots belong to k (see Sect.…”

“…Lemma (1.4) of [7]). U is a complete discrete valuation field of rank n with a pro-p-quasi-finite residue field k' such that ab t kp/k is a 77p-extension.…”