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arXiv · 2208.05007

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

Farshid Hajir·Christian Maire·Ravi Ramakrishna
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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.

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Farshid Hajir, Christian Maire, Ravi Ramakrishna. 2022-08-09. On tame ${\mathbb Z}/p{\mathbb Z}$ extensions with prescribed ramification. https://arxiv.org/abs/2208.05007

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