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Witt kernels of quadratic forms for multiquadratic extensions in characteristic 2

Author: Detlev W. Hoffmann
Journal: Proc. Amer. Math. Soc. 143 (2015), 5073-5082
MSC (2010): Primary 11E04; Secondary 11E81, 12F15
Published electronically: May 22, 2015
MathSciNet review: 3411127
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Abstract: Let $ F$ be a field of characteristic $ 2$ and let $ K/F$ be a purely inseparable extension of exponent $ 1$. We show that the extension is excellent for quadratic forms. Using the excellence, we recover and extend results by Aravire and Laghribi who computed generators for the kernel $ W_q(K/F)$ of the natural restriction map $ W_q(F)\to W_q(K)$ between the Witt groups of quadratic forms of $ F$ and $ K$, respectively, where $ K/F$ is a finite multiquadratic extension of separability degree at most $ 2$.

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  • [1] Roberto Aravire and Ahmed Laghribi, Results on Witt kernels of quadratic forms for multi-quadratic extensions, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4191-4197. MR 3105862,
  • [2] Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530 (2009d:11062)
  • [3] Richard Elman, T. Y. Lam, and Adrian R. Wadsworth, Amenable fields and Pfister extensions, Conference on Quadratic Forms--1976 (Proc. Conf., Queen's Univ., Kingston, Ont., 1976) Queen's Univ., Kingston, Ont., 1977, pp. 445-492. With an appendix ``Excellence of $ F(\varphi )/F$ for 2-fold Pfister forms'' by J. K. Arason. Queen's Papers in Pure and Appl. Math., No. 46. MR 0560497 (58 #27756)
  • [4] Detlev W. Hoffmann, Diagonal forms of degree $ p$ in characteristic $ p$, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 135-183. MR 2058673 (2005j:11028),
  • [5] Detlev W. Hoffmann, Witt kernels of bilinear forms for algebraic extensions in characteristic 2, Proc. Amer. Math. Soc. 134 (2006), no. 3, 645-652 (electronic). MR 2180880 (2006h:11045),
  • [6] Detlev W. Hoffmann and Ahmed Laghribi, Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4019-4053 (electronic). MR 2058517 (2005e:11041),
  • [7] Kazuya Kato, Symmetric bilinear forms, quadratic forms and Milnor $ K$-theory in characteristic two, Invent. Math. 66 (1982), no. 3, 493-510. MR 662605 (83i:10027),
  • [8] Ahmed Laghribi, Witt kernels of function field extensions in characteristic 2, J. Pure Appl. Algebra 199 (2005), no. 1-3, 167-182. MR 2134299 (2006c:11041),
  • [9] Pasquale Mammone and Remo Moresi, Formes quadratiques, algèbres à division et extensions multiquadratiques inséparables, Bull. Belg. Math. Soc. Simon Stevin 2 (1995), no. 3, 311-319 (French, with French summary). MR 1338463 (96j:11052)

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Additional Information

Detlev W. Hoffmann
Affiliation: Fakultät für Mathematik, Technische Universität Dortmund, 44221 Dortmund, Germany

Keywords: Quadratic form, bilinear form, Pfister form, Witt group, excellent extension, purely inseparable extension, exponent of an inseparable extension
Received by editor(s): March 7, 2014
Received by editor(s) in revised form: September 16, 2014
Published electronically: May 22, 2015
Additional Notes: The research on this paper was supported in part by the DFG Projekt “Annihilators and kernels in Kato’s cohomology in positive characteristic and in Witt groups in characteristic $2$”.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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