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

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 \(\mu_{2^ m}\).

MSC:

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

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