Galois module structure of the square root of the inverse different in even degree tame extensions of number fields

JOURNAL OF ALGEBRA

卷:468

Article

Caputo, Luca¹  Vinatier, Stephan¹ 

[1] Univ Limoges, CNRS, XLIM, DMI,UMR 7252, 123 Ave Albert Thomas, F-87060 Limoges, France

关键词: Tamely ramified extensions;    Square root of the inverse different;    Frohlich Group Determinants;    Stickelberger theorem;    Hasse-Davenport theorem;    Root numbers;    Binary tetrahedral Galois extension;   

DOI  :  10.1016/j.jalgebra.2016.06.035

来源: Elsevier

【 摘 要 】

Let G be a finite group and let N/E be a tamely ramified G-Galois extension of number fields whose inverse different C-N/E is a square. Let A(N/E) denote the square root of C-N/E. Then AN/E is a locally free Z[G]-module, which is in fact free provided N/E has odd order, as shown by Erez. Using M. Taylor's theorem, we can rephrase this result by saying that, when N/E has odd degree, the classes of A(N/E) and ON (the ring of integers of N) in Cl(Z[G]) are equal (and in fact both trivial). We show that the above equality of classes still holds when N/E has even order, assuming that N/E is locally abelian. This result is obtained through the study of the Frohlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Jacobi sums, together with the Hasse-Davenport formula, are involved in this study. Finally, when G is the binary tetrahedral group, we use our result in conjunction with Taylor's theorem to exhibit a tame G-Galois extension whose square root of the inverse different has nontrivial class in Cl(Z[G]). (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free