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Let $f$ be a polynomial of degree $n$ over a field $F$ of characteristic $p$. Suppose $f'=0$. Show that $p\mid n$ and that $f$ has at most $n/p$ distinct roots. I can't solve this question, any help is welcome.


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up vote 2 down vote accepted

$f'=0$ is equivalent to $f\in F[X^p]$, so $p\mid n$. Furthermore, $f\in F[X^p]$ implies $f(X)=g(X^p)$ where $\deg g=n/p$, so $f$ has at most $n/p$ distinct roots. (Note that $a\neq b$ implies $a^p\neq b^p$ in a field of characteristic $p>0$.)

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the problem is this polynomial is not irreducible, in my book (Bhattacharya) the polynomial has to be irreducible. Thanks for your answer – user42912 Dec 5 '12 at 2:28
calm down please, I'm just asking if we need or not this polynomial be irreducible. – user42912 Dec 5 '12 at 11:46

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