nLab square-free element

Idea

The generalization of square-free integer and square-free polynomial to arbitrary unique factorization domains.

Definition

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 $x^2 R$.

Examples

• 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 $\overline{F}[x]$.

Properties

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 $R$.

Square-free factorization

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 $r_i$

$$r = \prod_{i = 0}^{n} r_i^i$$

where each of the elements $r_i$ are pairwise coprime and not in the group of units.

See also

• square-free integer

• square-free polynomial