On tame -extensions with prescribed ramification

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