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Cyclotomic extensions of number fields. (English) Zbl 1056.11058

Let \(K\) be a number field, \(l\) a prime number, and \(\zeta_l\) a primitive \(l\)th root of unity. The paper is devoted to the cyclotomic extension \(K(\zeta_l)/K\), giving explicit formulae for the discriminant, conductor, and different of this extension. Fixing a prime ideal \(\mathfrak p\) of \(K\) above \(l\), let \(P\) be the product of all prime ideals of \(K(\zeta_l)\) above \(\mathfrak p\). For any \(0\leq a\leq b\) such that \(P^a/P^b\) is an \(\mathbb F_l\)-module (i.e., \(b-a\) is less than or equal to the absolute ramification index of a prime of \(K(\zeta_l)\) above \(\mathfrak p\)) the Galois module structure of \(P^a/P^b\) is obtained. For any field \(M\) such that \(K\subseteq M\subseteq K(\zeta_l)\), the ramification index and the residual degree of a prime of \(M\) above \(\mathfrak p\) are given explicitly. In addition, the Galois module structure of the unit group of \(K(\zeta_l)\) modulo \(l\)th powers is derived.

MSC:

11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
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References:

[1] Cohen, H.; Diaz y Diaz, F.; Olivier, M., Density des discriminants des extensions cycliques de degré premier, C.R. acad. sci. Paris, 330, 61-66, (2000) · Zbl 0941.11042
[2] Cohen, H.; Diaz y Diaz, F.; Olivier, M., On the density of discriminants of cyclic extensions of prime degree, J. reine angew. math., 550, 169-209, (2002) · Zbl 1004.11063
[3] Lang, S., Algebraic number theory, () · Zbl 0211.38404
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