Let $\displaystyle P$ be an arbitrary field, while $\displaystyle C$ is the complex field. $\displaystyle f(x)$ is a polynomial in $\displaystyle P$ with degree $\displaystyle >0$. Suppose $\displaystyle f$ has no multiple factor in $\displaystyle P$, prove then $\displaystyle f $ has no multiple root in $\displaystyle C.$

Here, we mean by $\displaystyle g(x)$ is a multiple factor of $\displaystyle f(x)$, if $\displaystyle g(x)$ is irreducible, and for some $\displaystyle k>1$, $\displaystyle g^k(x)\mid f(x).$