On tame -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.