The Witt ring kernel for a fourth degree field extension and related problems
type: Item
instance of: scholarly article
title: The Witt ring kernel for a fourth degree field extension and related problems
zbMATH Open document ID: 1226.11050
published in: Journal of Pure and Applied Algebra
publication date: 2009-11-04T00:00:00Z
review text: Let \(F\) be a field of characteristic not \(2\). An important question in the algebraic theory of quadratic forms is the determination of the kernel \(W(L/F)\) of the restriction map \(W(F)\to W(L)\) from the Witt ring of the field \(F\) to that of an extension field \(L\) over \(F\). It is known that \(W(L/F)=0\) for odd degree extensions due to the so-called Springer's theorem, and for \(L=F(\sqrt{d})\) the kernel is generated by the class of the norm form \(\langle 1,-d\rangle\) of the quadratic extension. In the present paper, the author determines \(W(L/F)\) in the case \([L:F]=4\) by explicitly describing the generators of this ideal. Previously, the kernel of such degree \(4\) extensions has only been determined in the situation where \(L\) contains a quadratic subextension of \(F\). It turns out that in general the kernel is generated by forms of type \(\langle 1,-d\rangle\) with \(F\subset F(\sqrt{d})\subset L\), and by \(2\)-fold Pfister forms \(\langle\!\langle p(\alpha),-\alpha\rangle\!\rangle\), where \(p(X)\in F[X]\) is the resolvent cubic of the minimal polynomial \(X^4+aX^2+bX+c\in F[X]\) of a suitable primitive element generating the extension \(L/F\). It is furthermore shown (invoking the Milnor conjecture proved by Voevodsky) that for such degree \(4\) extensions \(L/F\), one has \(W(L/F)\cap I^n(F)=(W(L/F)\cap I^2(F))I^{n-2}F\) for \(n\geq 2\), where \(I^n(F)\) denotes the \(n\)-th power of the fundamental ideal of classes of even-dimensional forms in \(W(F)\). The author also provides further applications of his results. He gives a necessary and sufficient criterion for when the tensor product \(Q\otimes_F B\) of a quaternion algebra \(Q\) and a biquaternion algebra \(B\) over \(F\) is not a division algebra. In the last section of the paper, the author studies degree \(4\) extensions \(L/F\) containing a quadratic subextension. The smallest Galois extension \(M\) of \(F\) containing such an \(L\) is either \(L\) itself, or (if \(L/F\) is not G…
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 5624920
zbMATH Keywords: quadratic form
zbMATH Keywords: isotropy
zbMATH Keywords: Witt ring
zbMATH Keywords: Witt kernel
zbMATH Keywords: quaternion algebra
zbMATH Keywords: biquaternion algebra
zbMATH Keywords: division algebra
zbMATH Keywords: exact sequence
zbMATH Keywords: Milnor conjecture
author: A. S. Sivatski
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.04.015
OpenAlex ID: W1975653415
cites work: Biquaternion algebras and quartic extensions
cites work: Q3708857
cites work: Witt Rings and Brauer Groups Under Multiquadratic Extensions, I
cites work: Design of Antireflection Grading Using Magneto-Dielectric Materials
cites work: Algebras with involution that become hyperbolic under a given extension
cites work: Motivic cohomology with \(\mathbb Z/2\)-coefficients
cites work: Q3960975
cites work: Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture
cites work: On the tensor product of division algebras
cites work: Generic splitting fields of central simple algebras: Galois cohomology and nonexcellence
cites work: Zero Divisors in Tensor Products of Division Algebras
cites work: Nonexcellence of the function field of the product of two conics
cites work: Some applications of Jordan norms to involutorial simple associative algebras
cites work: The Albert Quadratic Form for an Algebra of Degree Four
cites work: Quadratic forms under algebraic extensions
cites work: Q3859851
cites work: Q3516073
DOI: 10.1016/J.JPAA.2009.04.015
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description: scientific article; zbMATH DE number 5624920