# Properties of a polynomial with zero discriminant

In Wikipedia it says[1] that

"The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non-constant common divisor."

How does one prove this fact?

If $P(x)$ is a polynomial with degree $n$ with coefficients in an integer domain $D$, if $K$ is the algebraic closure of the ring of fractions of $D$ and if $r_1,\ldots,r_n\in K$ are the roots of $P(x)$ (there may be repeated roots, of course), then the discriminant $\Delta$ of $P(x)$ is $\left(\prod_{k=1}^n(r_i-r_j)\right)^2$. Therefore\begin{align}\Delta=0&\iff\text{there are repeated roots}\\&\iff\text{there are roots with degree >1}\\&\iff P(x)\text{ and }P'(x)\text{ have a common root.}\end{align}