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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