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Witt equivalence classes of quartic number fields. (English) Zbl 0742.11022
Two number fields are said to be Witt equivalent if their Witt rings of anisotropic quadratic forms are isomorphic. The classification of number fields of degree \(n\) over the rationals with respect to Witt equivalence amounts to (1) determining the (finite) number of Witt equivalence classes of fields of degree \(n\), (2) producing a list of representatives of all the classes, and (3) establishing criteria for an arbitrary field of degree \(n\) to belong to a given class. These problems have recently been solved for quadratic number fields [A. Czogala, Acta Arith. 58, 27-46 (1991; Zbl 0733.11012) and the reviewer, Math. Slovaca 41, 315- 330 (1991) and independently by J. Carpenter, Math. Z. 209, 153-166 (1992; Zbl 0724.11021)] and for cubic fields [the reviewer, Commun. Algebra 19, No. 4, 1125-1149 (1991; Zbl 0724.11020)]. The latter paper also solves (1) for \(n\leq 10\), showing in particular that there are exactly 29 Witt equivalence classes of quartic number fields. The present authors solve problems (2) and (3) for \(n=4\). They show that some properly chosen quadratic extensions of the fields \(\mathbb{Q}(\sqrt 2)\) and \(\mathbb{Q}(\sqrt 17)\) represent 26 Witt equivalence classes of quartic fields. They also find representatives for the remaining three classes consisting of quartic fields without quadratic subfields. The authors’ solution of problem (3) for \(n=4\) is divided into 11 cases depending on the splitting type of the principal ideal \(2O_ F\) in the field \(F\).

MSC:
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E08 Quadratic forms over local rings and fields
11R16 Cubic and quartic extensions
11E12 Quadratic forms over global rings and fields
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