Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 .

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.