Abstract.
In this paper we present a general view of the totally and wildly ramified extensions of degree p of a p-adic field K. Our method consists in deducing the properties of the set of all extensions of degree p of K from the study of the compositum \({\cal C}_K(p)\) of all its elements. We show that in fact \({\cal C}_K(p)\) is the maximal abelian extension of exponent p of F = F(K), where F is the compositum of all cyclic extensions of K of degree dividing p − 1. By our method, it is fairly simple to recover the distribution of the extensions of K of degree p (and also of their isomorphism classes) according to their discriminant.
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Del Corso, I., Dvornicich, R. The Compositum of Wild Extensions of Local Fields of Prime Degree. Mh Math 150, 271–288 (2007). https://doi.org/10.1007/s00605-006-0417-5
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DOI: https://doi.org/10.1007/s00605-006-0417-5