The quadratic discriminants and the Stickelberger congruence. (Les discriminants quadratiques et la congruence de Stickelberger.) (French) Zbl 0731.11061

Martinet, Jacques

Sémin. Théor. Nombres Bordx., Sér. II 1, No. 1, 197-204 (1989).

A classical theorem of Stickelberger asserts that the discriminant of an algebraic number field is congruent to \(0\) or \(1 \bmod 4\). In this paper, the following generalization is proved: let \(L/K\) be a finite extension of algebraic number fields, \(\Delta\) the relative discriminant, \(c\) the number of complex primes of \(L\) above real primes of \(K\), \(N=N_{K/{\mathbb Q}}\); then \((-1)^ cN(\Delta)\equiv 0\) or \(1 \bmod 4\); if \(K\) contains \(\mu_ 4\), then \(N(\Delta)\equiv 0, 1\) or \(4 \bmod 8\).
The proof proceeds by reduction to the case where \(L/K\) is a quadratic extension such that \(N(\Delta)\) is odd. In this case, it is shown, by using Hilbert symbols, that \(N(\Delta)\equiv (-1)^ c \bmod 2^{m+1}\) if \(K\) contains \(\mu_{2^ m}\).

Reviewer: T.Nguyen Quang Do (Besançon)

MSC:

11R29	Class numbers, class groups, discriminants
11R11	Quadratic extensions

Keywords:

Stickelberger theorem; discriminant; quadratic extension

References:

[1]	Bergé, A-M. et Martinet, J., Formes quadratiques et extensions en caractéristique 2, Ann. Inst. Fourier35, 2 (1985), 57-77. · Zbl 0539.10018
[2]	Frôhlich, A., Discriminants of algebraic number fields, Math. Z.74 (1960), 18-28. · Zbl 0098.03303
[3]	Frôhlich, A., Ideals in an extension field as modules over the algebraic integers in a finite number field, Math. Z.74 (1960), 29-38. · Zbl 0098.03401
[4]	Hecke, E., “Vorlesungen über die Theorie der algebraischen Zahlen,” Chelsea, New York, 1948 (éd. originale: 1923). · Zbl 0041.01102
[5]	Martinet, J., Character theory and Artin L-functions, “Algebraic Number Fields (A. Frôhlich, éd.),” Academic Press, London, New York, 1977, pp. 2-87. · Zbl 0359.12015
[6]	Martinet, J., Petits discriminants des corps de nombres, Series 56 (1982), 151-193. · Zbl 0491.12005
[7]	Schur, I., Elementarer Beweis eines Satzes von L. Stickelberger, Math. Z.29 (1929), 87-88 (= Ges. Abh., vol. III, pp. 87-88). · JFM 54.0190.01
[8]	Serre, J-P., Conducteurs d’Artin des caractères réels, Invent. Math.14 (1971), 173-183. · Zbl 0229.13006
[9]	Stickelberger, L., Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper, Int. Cong. Zürich (1897), 182-193. · JFM 29.0172.03
[10]	Weil, A., “Automorphic forms and Dirichlet Series,” 189, Heidelberg, 1971. · Zbl 0218.10046