Effective determination of the decomposition of the rational primes in a cubic field. (English) Zbl 0514.12003

Soit \(K\) un corps cubique de discriminant \(D\), \(K=\mathbb Q(\theta)\), \(\theta\) racine du polynôme irréductible \(f=X^3-aX+b\), \(a, b\in\mathbb Z\), et soit \(\Delta=(i(\theta))^2D\) le discriminant de \(f\). Pour tout nombre premier \(p\) (en distinguant les cas \(p=2\), \(p=3\) et \(p>3\)) les auteurs obtiennent la décomposition effective de \(p\mathbb Z\) dans \(K\) en fonction des coefficents du polynome \(f\); les auteurs en déduisent \(v_p(D)\) et retrouvent ainsi un théorème de H. Hasse sur les valeurs possibles du discriminant d’un corps cubique.
[Authors’ abstract: The decomposition of the rational primes in a cubic field \(K\) is determined in terms of the coefficients of a defining polynomial of \(K\). As a consequence, the discriminant \(D\) of \(K\) is straightforwardly computed and the cubic fields with index \(i(K) = 2\) are easily characterized.]
Reviewer: M.-N. Gras


11R16 Cubic and quartic extensions
11R29 Class numbers, class groups, discriminants
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