Cale Bases in Algebraic Orders

Picavet-L’Hermitte, Martine

Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 1, pp. 117-131.

Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline{R}$. 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 $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}\left(\overline{R}\right)\to \mathrm{Spec}\left(R\right)$ 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.

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