On tame ${\mathbb Z}/p{\mathbb Z}$ extensions with prescribed ramification

TL;DR

This paper provides a new, simpler proof of the tame Gras-Munnier Theorem, characterizing certain ${ m Z}/p{ m Z}$ extensions of number fields with prescribed ramification, and relates these extensions to dependence relations on Frobenius elements.

Contribution

It offers a novel, streamlined proof of the theorem and connects the set of extensions to dependence relations, enhancing understanding of ramification in number fields.

Findings

Simplified proof of the Gras-Munnier Theorem

Relation between extensions and Frobenius dependence relations

Reproof of key proposition using Wiles-Greenberg formula

Abstract

The tame Gras-Munnier Theorem gives a criterion for the existence of a ${\mathbb Z}/{\mathbb Z}$-extension of a number field $K$ ramified at exactly a set $S$ of places of $K$ prime to $p$ (allowing real Archimedean places when $p=2$) in terms of the existence of a dependence relation on the Frobenius elements of these places in a certain governing extension. We give a new and simpler proof of this theorem that also relates the set of such extensions of $K$ to the set of these dependence relations. After presenting this proof, we then reprove the key Proposition 3 using the more sophisticated Wiles-Greenberg formula based on global duality.