An analogue of Pfister's local-global principle in the burnside ring

Martin Epkenhans

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 1, page 31-44
  • ISSN: 1246-7405

Abstract

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Let N / K be a Galois extension with Galois group 𝒢 . We study the set 𝒯 ( 𝒢 ) of ℤ -linear combinations of characters in the Burnside ring ℬ ( 𝒢 ) which give rise to ℤ -linear combinations of trace forms of subextensions of N / K which are trivial in the Witt ring W ( K ) of K . In particular, we prove that the torsion subgroup of ℬ ( 𝒢 ) / 𝒯 ( 𝒢 ) coincides with the kernel of the total signature homomorphism.

How to cite

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Epkenhans, Martin. "An analogue of Pfister's local-global principle in the burnside ring." Journal de théorie des nombres de Bordeaux 11.1 (1999): 31-44. <http://eudml.org/doc/248342>.

@article{Epkenhans1999,
abstract = {Let $N/K$ be a Galois extension with Galois group $\mathcal \{G\}$. We study the set $\mathcal \{T\}(\mathcal \{G\})$ of $\mathbb \{Z\}$-linear combinations of characters in the Burnside ring $\mathcal \{B\}(\mathcal \{G\})$ which give rise to $\mathbb \{Z\}$-linear combinations of trace forms of subextensions of $N/K$ which are trivial in the Witt ring W$(K)$ of $K$. In particular, we prove that the torsion subgroup of $\mathcal \{B\}(\mathcal \{G\}) / \mathcal \{T\}(\mathcal \{G\})$ coincides with the kernel of the total signature homomorphism.},
author = {Epkenhans, Martin},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {trace forms; Burnside ring; Witt ring},
language = {eng},
number = {1},
pages = {31-44},
publisher = {Université Bordeaux I},
title = {An analogue of Pfister's local-global principle in the burnside ring},
url = {http://eudml.org/doc/248342},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Epkenhans, Martin
TI - An analogue of Pfister's local-global principle in the burnside ring
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 1
SP - 31
EP - 44
AB - Let $N/K$ be a Galois extension with Galois group $\mathcal {G}$. We study the set $\mathcal {T}(\mathcal {G})$ of $\mathbb {Z}$-linear combinations of characters in the Burnside ring $\mathcal {B}(\mathcal {G})$ which give rise to $\mathbb {Z}$-linear combinations of trace forms of subextensions of $N/K$ which are trivial in the Witt ring W$(K)$ of $K$. In particular, we prove that the torsion subgroup of $\mathcal {B}(\mathcal {G}) / \mathcal {T}(\mathcal {G})$ coincides with the kernel of the total signature homomorphism.
LA - eng
KW - trace forms; Burnside ring; Witt ring
UR - http://eudml.org/doc/248342
ER -

References

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  1. [1] P. Beaulieu and T. Palfrey.The Galois number. Math. Ann.309 (1997), 81-96. Zbl0885.11027MR1467647
  2. [2] P.E. Conner and R. Perlis.A Survey of Trace Forms of Algebraic Number Fields. World Scientific, Singapore, (1984). Zbl0551.10017MR761569
  3. [3] C. Drees, M. Epkenhans, and M. Krüskemper.On the computation of the trace form of some Galois extensions. J. Algebra, 192 (1997), 209-234. Zbl0873.11027MR1449959
  4. [4] M. Epkenhans and M. Krüskemper.On Trace Forms of étale Algebras and Field Extensions. Math. Z.217 (1994), 421-434. Zbl0821.11024MR1306669
  5. [5] W. Scharlau.Quadratic and Hermitian Forms. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, (1985). Zbl0584.10010MR770063
  6. [6] T.A. Springer.On the equivalence of quadratic forms. Proc. Acad. Amsterdam, 62 (1959), 241-253. Zbl0087.03501MR108468
  7. [7] O. Taussky.The Discriminant Matrices of an Algebraic Number Field. J. London Math. Soc.43 (1968), 152-154. Zbl0155.37903MR228473

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