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${\mathbb {Z}}/p{\mathbb {Z}}$-extensions with prescribed ramification
The tame Gras–Munnier Theorem gives a criterion for the existence of a 
$ {\mathbb Z}/p{\mathbb Z} $-extension of a number field K ramified at exactly a tame set S of places of K, the finite 
$v \in S$ necessarily having norm 
$1$ mod p. The criterion is the existence of a nontrivial dependence relation on the Frobenius elements of these places in a certain governing extension. We give a short new proof which extends the theorem by showing the subset of elements of 
$H^1(G_S,{\mathbb {Z}}/p{\mathbb {Z}})$ giving rise to such extensions of K has the same cardinality as the set of these dependence relations. We then reprove the key Proposition 2.2 using the more sophisticated Greenberg–Wiles formula based on global duality.
