Constructing class fields over local fields

Sebastian Pauli[1]

  • [1] Department of Mathematics and Statistics University of North Carolina Greensboro Greensboro, NC 27402, USA

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 3, page 627-652
  • ISSN: 1246-7405

Abstract

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Let K be a 𝔭 -adic field. We give an explicit characterization of the abelian extensions of K of degree p by relating the coefficients of the generating polynomials of extensions L / K of degree p to the exponents of generators of the norm group N L / K ( L * ) . This is applied in an algorithm for the construction of class fields of degree p m , which yields an algorithm for the computation of class fields in general.

How to cite

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Pauli, Sebastian. "Constructing class fields over local fields." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 627-652. <http://eudml.org/doc/249641>.

@article{Pauli2006,
abstract = {Let $K$ be a $\{\mathfrak\{p\}\}$-adic field. We give an explicit characterization of the abelian extensions of $K$ of degree $p$ by relating the coefficients of the generating polynomials of extensions $L/K$ of degree $p$ to the exponents of generators of the norm group $N_\{L/K\}(L^*)$. This is applied in an algorithm for the construction of class fields of degree $p^m$, which yields an algorithm for the computation of class fields in general.},
affiliation = {Department of Mathematics and Statistics University of North Carolina Greensboro Greensboro, NC 27402, USA},
author = {Pauli, Sebastian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {abelian extensions of local fields},
language = {eng},
number = {3},
pages = {627-652},
publisher = {Université Bordeaux 1},
title = {Constructing class fields over local fields},
url = {http://eudml.org/doc/249641},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Pauli, Sebastian
TI - Constructing class fields over local fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 627
EP - 652
AB - Let $K$ be a ${\mathfrak{p}}$-adic field. We give an explicit characterization of the abelian extensions of $K$ of degree $p$ by relating the coefficients of the generating polynomials of extensions $L/K$ of degree $p$ to the exponents of generators of the norm group $N_{L/K}(L^*)$. This is applied in an algorithm for the construction of class fields of degree $p^m$, which yields an algorithm for the computation of class fields in general.
LA - eng
KW - abelian extensions of local fields
UR - http://eudml.org/doc/249641
ER -

References

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  10. R. E. MacKenzie, G. Whaples, Artin-Schreier equations in characteristic zero. Amer. J. Math. 78 (1956), 473–485. MR 19,834c Zbl0073.26402MR90584
  11. P. Panayi, Computation of Leopoldt’s p-adic regulator. Dissertation, University of East Anglia, 1995. 
  12. S. Pauli, X.-F. Roblot, On the computation of all extensions of a p-adic field of a given degree. Math. Comp. 70 (2001), 1641–1659. Zbl0981.11038MR1836924
  13. J.-P. Serre, Corps locaux. Hermann, Paris, 1963. Zbl0137.02601MR354618
  14. K. Yamamoto, Isomorphism theorem in the local class field theory. Mem. Fac. Sci. Kyushu Ser. A 12 (1958), 67–103. Zbl0083.25901MR150136

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