## 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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