Computing class fields via the Artin map

Abstract

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. PreliminariesLet k/Q be an algebraic number field; we denote its ring of integers by o k . A congruence module m formally consists of an integral ideal m 0 and a formal product m ∞ of real places (.) (i) of k viewed as embeddings into R. By I m we denote the set of all ideals coprime to m 0 , by P m the set of principal ideals generated by elements α ≡ 1 mod * m (i.e., α ≡ 1 mod m 0 and α (i) > 0 for all (.) (i) |m ∞ ) and finally by Cl m := I m /P m 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 I m containing P m . For any ideal group H letH := I m /H.

The distribution of second p-class groups on coclass graphs

“…Further, we gratefully acknowledge helpful advice for constructing class fields [36] with the aid of MAGMA [56,18,19] by Claus Fieker, University of Kaiserslautern.…”

Computational Class Field Theory

“…Our earlier observation that we may decompose I f /A f ∼ = Gal(L/K) into a product of cyclic groups of prime power order and generate L accordingly as a compositum of cyclic extensions of K is particularly relevant in this context, as it reduces our problem to a number of instances where K ⊂ L is cyclic of prime power degree. Current implementations [10] deal with prime power values up to 20.…”

Index-<i>p</i> Abelianization Data of <i>p</i>-Class Tower Groups

“…Based on Fieker's technique [40], we use the computational algebra system MAGMA [3] [4] to construct these extensions and to calculate their arithmetical invariants. In Table 2, which is continued in Table 3 on the following page, we present the kernel i  of the 3-principalization of K in i L [21] [23], the occupation numbers ( ) i o  of the principalization kernels [19], and the abelian type invariants i τ , resp.…”