The generalization of square-free integer and square-free polynomial to arbitrary unique factorization domains.
Let $R$ be a unique factorization domain. An element $r \in R$ is square-free if for all irreducible elements $x \in R$, $r$ is not in the ideal $x^2 R$.
A square-free integer is a square-free element in the integers $\mathbb{Z}$.
Let $F$ be a discrete field and let $\overline{F}$ be the algebraic closure of $F$. A univariate square-free polynomial with coefficients in $\overline{F}$ is a square-free element in the polynomial ring $\overline{F}[x]$.
Given a unique factorization domain $R$ such that for every irreducible element $p \in R$, the ideal $p R$ is a maximal ideal, and a square-free element $x \in R$, the quotient ring $R/x R$ is a reduced ring. Since for every non-zero element $r \in R$, $R/x R$ is a prefield ring, $R/x R$ is an integral domain and thus a field if and only if $x$ is an irreducible element in $R$.
Given a non-zero element of a unique factorization domain $r \in R$, the square-free factorization or square-free decomposition of $r$ is a factorization of $r$ into powers of square-free elements $r_i$
where each of the elements $r_i$ are pairwise coprime and not in the group of units.
Last revised on January 22, 2023 at 20:45:31. See the history of this page for a list of all contributions to it.