Cale Bases in Algebraic Orders

Abstract

Cale Bases in Algebraic OrdersMartine Picavet-L'Hermitte AbstractLet R be a non-maximal order in a finite algebraic number field with integral closure R. Although R is not a unique factorization domain, we obtain a positive integer N and a family Q (called a Cale basis) of primary irreducible elements of R such that x N has a unique factorization into elements of Q for each x ∈ R coprime with the conductor of R. Moreover, this property holds for each nonzero x ∈ R when the natural map Spec(R) → Spec(R) is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.