
Multiple root
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 $\displaystyle k>1$, $\displaystyle g^k(x)\mid f(x).$

Re: Multiple root
This doesn't make any sense as written? I'm going to assume that $\displaystyle P$ is a subfield of $\displaystyle \mathbb{C}$. Recall then that $\displaystyle f$ having no multiple roots is equivalent to $\displaystyle (f,f')=1$, which is trivially independent of what field you are discussing. Once again, this question dreally doesn't make sense.