Longueurs des décompositions en produits d'éléments irréductibles dans un anneau de Dedekind. (Lengths of decompositions in products of irreducible elements in a Dedekind ring) (Q1076087)

Let A be a Dedekind domain and C(A) be its group of ideal classes. It is shown that there exists a constant \(q\geq 1\) such that A satisfies \(E_ q\), namely, for any two finite families \(a_ 1,...,a_ r\) and \(b_ 1,..,b_ s\) of irreducible (extremal) elements of A, if the ideal generated by the a's is equal to the ideal generated by the b's, then 1/q\(\leq r/s\leq q\). If we define \(\ell (C(A))\) as the smallest integer m such that for each family of m elements of C(A) we can extract a subfamily whose sum is zero, then A satisfies \(E_ q\) with \(q=\ell (C(A))/2\). This bound is attained when each non-principal class of ideals of A contains at least one prime ideal of A. In \(propositions\quad 3,\) it is shown hat when C(A) is isomorphic to the direct sum of cyclic subgroups generated by the elements of \(P_ A\) which is the set of ideal classes containing at least one non-principal ideal, then A satisfies \(E_ q\) with \(q=m/2\). Here m is the largest of the orders of elements of \(P_ A\) and q is the best possible. The results above are used to get information on the number and invertibility of the roots of some families of polynomials over A and to get a lower bound for the number of ideal classes when A is the ring of integers of an imaginary quadratic field.