Laura Capuano^[1] and Ilaria Del Corso^[2]

A note on upper ramification jumps in Abelian extensions of exponent p

Pages: 317-329
Received: 30 July 2015   
Accepted in revised form: 16 October 2015
Mathematics Subject Classification (2010): 11S15, 11S20, 11S31.
Keywords:Elementary Abelian p-extensions, upper ramification jumps, normic groups, class field theory.
Author address:
[1] : Scuola Normale Superiore, Piazza dei Cavalieri 7, 56127 Pisa, Italy
[2] : Università di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

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Abstract: In this paper we present a classification of the possible upper ramification jumps for an elementary Abelian \(p\)-extension of a \(p\)-adic field. The fundamental step for the proof of the main result is the computation of the ramification filtration for the maximal elementary Abelian \(p\)-extension of the base field \(K\). This result generalizes [3, Lemma 9, p.286], where the same result is proved under the assumption that \(K\) contains a primitive \(p\)-th root of unity. To deal with this general case we use class field theory and the explicit relations between the normic group of an extension and its ramification jumps, and we obtain necessary and sufficient conditions for the upper ramification jumps of an elementary Abelian \(p\)-extension of \(K\).

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