hello everyone, this is my first post.. I've got myself a problem.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Toughie about multiple roots

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