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On the computation of the trace form of some Galois extensions. (English) Zbl 0873.11027

This paper investigates the trace form tr\(_{L/K}: L \to K : x \to \) tr\(_{L/K}x^2\) of a finite Galois extension \(L/K\). The isometry class of the trace form is determined for most, but not all, Galois extensions of degree \(\leq 31\). In particular, the trace of a cyclic extension of degree 16 and of an extension with Galois group \(Q_{16}\), the generalized quaternion group of order 16, are not computed, but trace forms for Galois extensions for all other groups of order 16 are obtained. Finally the trace form of a cyclotomic extension of \({\mathbb{Q}}\) and of its maximal real subfield are determined. The trace form of a cyclotomic extension has already been computed by P. E. Conner and R. Perlis [A survey of trace forms of algebraic number fields (World Scientific, Singapore) (1984; Zbl 0551.10017)], but the proof given here is shorter.

MSC:

11E12 Quadratic forms over global rings and fields
11R32 Galois theory

Citations:

Zbl 0551.10017
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Full Text: DOI

References:

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