# Contents

## Idea

Given a discrete field $F$, let $\overline{F}$ denote its algebraic closure. A square-free polynomial is a univariate polynomial $q \in \overline{F}[x]$ such that for all elements $a \in \overline{F}$, $(x - a)^2$ does not divide $q$.

## Properties

Given a discrete field $F$, let $\overline{F}$ denote its algebraic closure. Given a square-free polynomial $q \in \overline{F}[x]$, the quotient ring $\overline{F}[x]/q\overline{F}[x]$ is a reduced ring. Since for every polynomial $q \in \overline{F}[x]$, $\overline{F}[x]/q\overline{F}[x]$ is a prefield ring, $\overline{F}[x]/q\overline{F}[x]$ is an integral domain and thus a field if and only if $q$ is a prime polynomial in $\overline{F}[x]$, a monic polynomial of degree one; the resulting quotient ring is equivalent to $\overline{F}$.