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If a square free function $ f \in K[x] $ , where $\mathbb K$ is a field of characterstic $0$ . How can i show that the root of $f$ ( in the splitting field ) are distinct.

What more possibly can i deduce from this proof ? anything about the degree of root of $f$ ? Thank you for you help .

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1 Answer 1

If $\alpha$ is a multiple root of $f$, then $\alpha$ is also a root of $f'$. Hence both $f$ and $f'$ are multiple of the minimal polynomial $p$ of $\alpha$. Write $f=pg$. Then $f'=p'g+pg'$. Hence $p$ divides $p'g$ and since $p$ is irreducible and $p'$ is a nonzero polynomial of lesser degree, we see that $p$ is relatively prime to $p'$, hence $p$ divides $g$. But then $p^2$ divides $f$.

Note that characteristic $0$ is used when we conlcude that $p'$ is nonzero.

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