Parity of class numbers and Witt equivalence of quartic fields.

*(English)*Zbl 0854.11023
Math. Comput. 64, No. 212, 1711-1715 (1995); Corrigendum ibid. 66, No. 218, 927 (1997).

The authors first prove the following result, the statement of which they attribute to P. E. Conner: Let \(F\) be an algebraic number field such that \(-1\) is a sum of two squares in \(F\) but not itself a square in \(F\), and such that \(-1\) is a square in every one of the dyadic completions of \(F\) (i.e., the level of \(F\) is 2 and each dyadic completion of \(F\) has level 1). Then every number field which is Witt equivalent to \(F\) has even ideal class number.

The remainder of the paper concerns the converse question: If every number field Witt equivalent to \(F\) has even ideal class number, does \(F\) necessarily satisfy the above conditions on levels? It follows from prior work of the last author [Commun. Algebra 19, 1125-1149 (1991; Zbl 0724.11020)] that this question has an affirmative answer for the cases of quadratic and cubic fields. In the present paper, the authors proceed to show that the answer is affirmative also for the case of quartic fields. It is known that there are exactly 29 Witt equivalence classes of such fields, and representative fields in each of these classes have been found previously by the first two authors [Math. Comput. 58, 355-368 (1992; Zbl 0742.11022)]. It is shown that the above conditions on levels are satisfied by the fields in precisely two of these 29 classes. The result is then established by finding a representative field in each of the remaining 27 classes which has odd class number (in fact, class number 1).

The remainder of the paper concerns the converse question: If every number field Witt equivalent to \(F\) has even ideal class number, does \(F\) necessarily satisfy the above conditions on levels? It follows from prior work of the last author [Commun. Algebra 19, 1125-1149 (1991; Zbl 0724.11020)] that this question has an affirmative answer for the cases of quadratic and cubic fields. In the present paper, the authors proceed to show that the answer is affirmative also for the case of quartic fields. It is known that there are exactly 29 Witt equivalence classes of such fields, and representative fields in each of these classes have been found previously by the first two authors [Math. Comput. 58, 355-368 (1992; Zbl 0742.11022)]. It is shown that the above conditions on levels are satisfied by the fields in precisely two of these 29 classes. The result is then established by finding a representative field in each of the remaining 27 classes which has odd class number (in fact, class number 1).

Reviewer: A.G.Earnest (Carbondale)

##### MSC:

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

11R29 | Class numbers, class groups, discriminants |

11R16 | Cubic and quartic extensions |

11E41 | Class numbers of quadratic and Hermitian forms |

##### Keywords:

class number parity; even ideal class number; levels; quartic fields; Witt equivalence classes
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\textit{S. Jakubec} et al., Math. Comput. 64, No. 212, 1711--1715 (1995; Zbl 0854.11023)

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##### References:

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[5] | R. Perlis, K. Szymiczek, P. E. Conner, and R. Litherland, Matching Witts with global fields, Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991) Contemp. Math., vol. 155, Amer. Math. Soc., Providence, RI, 1994, pp. 365 – 387. · Zbl 0807.11024 |

[6] | Kazimierz Szymiczek, Witt equivalence of global fields, Comm. Algebra 19 (1991), no. 4, 1125 – 1149. · Zbl 0724.11020 |

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