Factorization in almost Dedekind domain

Abstract

Let $F$ be a field, $p$ a prime number, $X$ an indeterminate over $F$, $D_n =F[X^{\frac{1}{p^n}}, X^{-\frac{1}{p^n}}]$ for each integer $n \geq 0$ and $D = \bigcup\limits_{n\in\mathbb{N}_0}D_n.$ Then $D$ is a one-dimensional B{é}zout domain but not a Dedekind domain, and $D$ is an almost Dedekind domain if and only if char$(F) \neq p$. In this paper, we study the element-wise factorization properties of $D$. For example, we determine when an irreducible element of $D_n$ is an irreducible element of $D$, in terms of $n$ and $p$. In particular, we show that if $F$ is algebraically closed or a finite field of char$(F)=p$, then $D$ has no irreducible element. We also show that if $F$ is a finite field of odd characteristic, then an irreducible element $f(X)$ of $D_0$ is irreducible in $D$ if and only if it is a factor of a cyclotomic polynomial $Φ_n(X)$ for some integer $n \geq 1$ which satisfies a certain equation in terms of $|F|$ and deg$(f(X))$. Finally, we introduce the notion of infinite product and we then show that if $F= \mathbb{Q}$ and $p=2$, every nonzero nonunit of $D$ can be written as a product of countably many prime elements of $D$ and every proper nonzero principal ideal of $D$ can be uniquely written as a countable intersection of principal primary ideals.