A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Nigel P. Byott[1]

  • [1] Mathematics Research Institute University of Exeter Harrison Building North Park Road Exeter EX4 4QF, UK

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 1, page 59-70
  • ISSN: 1246-7405

Abstract

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Let S / R be a finite extension of discrete valuation rings of characteristic p > 0 , and suppose that the corresponding extension L / K of fields of fractions is separable and is H -Galois for some K -Hopf algebra H . Let 𝔻 S / R be the different of S / R . We show that if S / R is totally ramified and its degree n is a power of p , then any element ρ of L with v L ( ρ ) ≡ - v L ( 𝔻 S / R ) - 1 ( mod n ) generates L as an H -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

How to cite

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Byott, Nigel P.. "A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 59-70. <http://eudml.org/doc/219828>.

@article{Byott2011,
abstract = {Let $S/R$ be a finite extension of discrete valuation rings of characteristic $p&gt;0$, and suppose that the corresponding extension $L/K$ of fields of fractions is separable and is $H$-Galois for some $K$-Hopf algebra $H$. Let $\mathbb\{D\}_\{S/R\}$ be the different of $S/R$. We show that if $S/R$ is totally ramified and its degree $n$ is a power of $p$, then any element $\rho $ of $L$ with $v_L(\rho ) \equiv -v_L(\mathbb\{D\}_\{S/R\})-1 \hspace\{4.44443pt\}(\@mod \; n)$ generates $L$ as an $H$-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.},
affiliation = {Mathematics Research Institute University of Exeter Harrison Building North Park Road Exeter EX4 4QF, UK},
author = {Byott, Nigel P.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Normal basis; Hopf-Galois extensions; local fields; normal basis},
language = {eng},
month = {3},
number = {1},
pages = {59-70},
publisher = {Société Arithmétique de Bordeaux},
title = {A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$},
url = {http://eudml.org/doc/219828},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Byott, Nigel P.
TI - A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 59
EP - 70
AB - Let $S/R$ be a finite extension of discrete valuation rings of characteristic $p&gt;0$, and suppose that the corresponding extension $L/K$ of fields of fractions is separable and is $H$-Galois for some $K$-Hopf algebra $H$. Let $\mathbb{D}_{S/R}$ be the different of $S/R$. We show that if $S/R$ is totally ramified and its degree $n$ is a power of $p$, then any element $\rho $ of $L$ with $v_L(\rho ) \equiv -v_L(\mathbb{D}_{S/R})-1 \hspace{4.44443pt}(\@mod \; n)$ generates $L$ as an $H$-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.
LA - eng
KW - Normal basis; Hopf-Galois extensions; local fields; normal basis
UR - http://eudml.org/doc/219828
ER -

References

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  1. N. P. Byott, Integral Hopf-Galois structures on degree p 2 extensions of p -adic fields. J. Algebra 248 (2002), 334–365. Zbl0992.11065MR1879021
  2. N. P. Byott and G. G. Elder, A valuation criterion for normal bases in elementary abelian extensions. Bull. London Math. Soc. 39 (2007), 705–708. Zbl1128.11055MR2365217
  3. L. N. Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory. Mathematical Surveys and Monographs 80, American Mathematical Society, 2000. Zbl0944.11038MR1767499
  4. G. G. Elder, A valuation criterion for normal basis generators in local fields of characteristic p . Arch. Math. 94 (2010), 43–47. Zbl1220.11143MR2581332
  5. C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions. J. Algebra 106 (1987), 239–258. Zbl0615.12026MR878476
  6. J.-P. Serre, Local Fields. Graduate Texts in Mathematics 67, Springer, 1979. Zbl0423.12016MR554237
  7. J.-P. Serre, Linear Representations of Finite Groups. Graduate Texts in Mathematics 42, Springer, 1977. Zbl0355.20006MR450380
  8. H. Stichtenoth, Algebraic Function Fields and Codes. Springer, 1993. Zbl1155.14022MR1251961
  9. L. Thomas, A valuation criterion for normal basis generators in equal positive characteristic. J. Algebra 320 (2008), 3811–3820. Zbl1207.11110MR2457723

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