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Witt kernels of bi-quadratic extensions in characteristic 2. (English) Zbl 1053.11032

The Witt kernel of a field extension \(L/k\) is the kernel of the homomorphism \(W(L) \rightarrow W(k)\) between the Witt groups of quadratic forms over \(L\) and \(k\). The first result of the paper asserts that for \(k\) of characteristic \(2\), any quadratic extension \(L\) of \(k\) is excellent, that is, the anisotropic part over \(L\) of any anisotropic quadratic \(k-\)form is defined over \(k\). For this the author uses his earlier results [Arch. Math. 63, 23–29 (1994, Zbl 0812.11027)]. The main result of the paper is the determination of the Witt kernel for biquadratic extensions of fields of characteristic \(2\). This is done separately for separable and inseparable extensions. The characterization of the forms in the Witt kernel is up to Witt equivalence and an example is given to show that this cannot be strengthened to isometry.

MSC:

11E04 Quadratic forms over general fields
12F05 Algebraic field extensions

Citations:

Zbl 0812.11027
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References:

[1] Scharlau, Quadratic and hermitian forms (1985) · doi:10.1007/978-3-642-69971-9
[2] Lam, The algebraic theory of quadratic forms (1973) · Zbl 0259.10019
[3] DOI: 10.1007/BF01196294 · Zbl 0812.11027 · doi:10.1007/BF01196294
[4] DOI: 10.1007/BF01189354 · Zbl 0263.15015 · doi:10.1007/BF01189354
[5] Baeza, Quadratic forms over semilocal rings 655 (1978) · doi:10.1007/BFb0070341
[6] Elman, Conf. Quad. Forms, 1976 46 pp 445– (1977)
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