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

Martin Epkenhans

doi:10.5802/jtnb.237

Martin Epkenhans

Journal de Théorie des Nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 31-44.

Résumé
Abstract

Soit $N / K$ une extension galoisienne de groupe de Galois $𝒢$ . On étudie l’ensemble $𝒯 (𝒢)$ des combinaisons linéaires sur $ℤ$ de caractères de l’anneau de Burnside $ℬ (𝒢)$ , qui induisent des combinaisons $ℤ$ -linéaires des formes trace de sous-extensions de $N / K$ qui sont triviales dans l’anneau de Witt W $(K)$ de $K$ . On montre que le sous-groupe de torsion de $ℬ (𝒢) / 𝒯 (𝒢)$ est le noyau de l’homomorphisme signature.

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.

MR 1730431 | Zbl 0964.11021

DOI : https://doi.org/10.5802/jtnb.237

@article{JTNB_1999__11_1_31_0,
     author = {Epkenhans, Martin},
     title = {An analogue of {Pfister's} local-global principle in the burnside ring},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {31--44},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {1},
     year = {1999},
     doi = {10.5802/jtnb.237},
     zbl = {0964.11021},
     mrnumber = {1730431},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.237/}
}

Martin Epkenhans. An analogue of Pfister's local-global principle in the burnside ring. Journal de Théorie des Nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 31-44. doi : 10.5802/jtnb.237. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.237/

Bibliographie

[1] P. Beaulieu and T. Palfrey. The Galois number. Math. Ann. 309 (1997), 81-96. | MR 1467647 | Zbl 0885.11027

[2] P.E. Conner and R. Perlis. A Survey of Trace Forms of Algebraic Number Fields. World Scientific, Singapore, (1984). | MR 761569 | Zbl 0551.10017

[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. | MR 1449959 | Zbl 0873.11027

[4] M. Epkenhans and M. Krüskemper. On Trace Forms of étale Algebras and Field Extensions. Math. Z. 217 (1994), 421-434. | MR 1306669 | Zbl 0821.11024

[5] W. Scharlau. Quadratic and Hermitian Forms. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, (1985). | MR 770063 | Zbl 0584.10010

[6] T.A. Springer. On the equivalence of quadratic forms. Proc. Acad. Amsterdam, 62 (1959), 241-253. | MR 108468 | Zbl 0087.03501

[7] O. Taussky. The Discriminant Matrices of an Algebraic Number Field. J. London Math. Soc. 43 (1968), 152-154. | MR 228473 | Zbl 0155.37903