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On the computation of all extensions of a $p$-adic field of a given degree


Authors: Sebastian Pauli and Xavier-François Roblot
Journal: Math. Comp. 70 (2001), 1641-1659
MSC (2000): Primary 11S15, 11S05; Secondary 11Y40
DOI: https://doi.org/10.1090/S0025-5718-01-01306-0
Published electronically: March 8, 2001
MathSciNet review: 1836924
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Abstract:

Let $\mathbf{k}$ be a $p$-adic field. It is well-known that $\mathbf{k}$ has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions $\mathbf{K}/\mathbf{k}$ of a given degree and discriminant.


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Additional Information

Sebastian Pauli
Affiliation: Centre Interuniversitaire en Calcul Mathématique Algébrique, Concordia University, 1455 de Maisonneuve Blvd. W., Montréal, Québec, H3G 1M8, CANADA
Email: pauli@cicma.concordia.ca

Xavier-François Roblot
Affiliation: Institut Girard Desargues, Université Claude Bernard (Lyon 1), 43, boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France
Email: roblot@desargues.univ-lyon1.fr

DOI: https://doi.org/10.1090/S0025-5718-01-01306-0
Keywords: $p$-adic fields, wildly ramified extensions, Eisenstein polynomials
Received by editor(s): May 24, 1999
Received by editor(s) in revised form: January 14, 2000
Published electronically: March 8, 2001
Additional Notes: The work of the first author was supported in part by ISM and FCAR/CICMA (Québec).
The work of the second author was supported in part by NSERC (Canada) and FCAR/CICMA (Québec).
We would like to thank David Ford for his careful reading of the original manuscript and for his useful comments.
Article copyright: © Copyright 2001 American Mathematical Society

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