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

The tame Gras-Munnier Theorem gives a criterion for the existence of a $\mathbb{Z}\mathrm{/}\mathbb{Z}$-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.