hello everyone, this is my first post.. I've got myself a problem. say I have a field F, and that f(x) is a polynomial with coefficients in F. (all the normal differentiation rules apply like always.. product rule, quotient rule, etc.) Now, I know that a polynomial f(x) has a "multiple root" if there exists some field E (containing F) and some element, alpha, of E such that (x-alpha)^2 divides f(x) in E[x]. my questoin is: how can I prove that a non-constant polynomial f(x) has a multiple root if and only if f(x) is not relatively prime to its derivative? Thank you to anyone who even takes a look at this.