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Article electronically published on March 24, 2000 COMPUTING CLASS FIELDS VIA THE ARTIN MAP

By Claus Fieker


Abstract. Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. We conclude with several examples. 1. Preliminaries Let k/Q be an algebraic number field; we denote its ring of integers by ok. A congruence module m formally consists of an integral ideal m0 and a formal product m ∞ of real places (.)(i) of k viewed as embeddings into R. ByImwe denote the set of all ideals coprime to m0, byPm the set of principal ideals generated by elements α ≡ 1mod ∗ m (i.e., α ≡ 1modm0 and α (i)> 0 for all (.)(i)|m∞) and finally by Clm: = Im /Pm the ray class group modulo m. There are efficient algorithms for computing ray class groups [6, 17] provided we already know the unit and class group of k. An ideal group H (defined mod m) is a subgroup of Im containing Pm. For any ideal group H let ¯ H: = Im /H. Now, let K/k be a finite extension. By dK/k we denote the relative discriminant of K/k as an ideal of ok. From now on K/k will always be a finite abelian extension. In this context we denote by σp the Frobenius automorphism belonging to the unramified prime ideal p of k. We will make extensive use of the Artin map, the multiplicative extension of the function mapping unramified prime ideals to their Frobenius automorphism: (., K/k):I d ∏ K/k → Gal(K/k):a = p vp(a) ↦ → ∏ where σp(x) ≡ x N(p) mod P for any P|p and any x ∈ oK. With this in mind we can define class fields: p|a p|a σ vp(a) p, Definition 1.1. Let m be a congruence module, H be an ideal group and K/k be an abelian extension. K is the class field belonging to H iff Gal(K/k) (.,K/k) ≃ Im /H. Received by the editor April 6, 1999 and, in revised form, August 16, 1999

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