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Let $h(x)$ be a polynomial of non-zero degree over a field $F$.

I am wondering whether there must always exist a Frobenius Polynomial $f \in F[x]$ satisfying the following:

  1. $\operatorname{deg}(f)>0$

  2. The geometric and algebraic multiplicities of each root of $f$ is $1$.

  3. The greatest common divisor $\gcd(f,f')=h(x)$.

Thank you for you help

math110
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  • Could you add more text to explain what you mean? Or could you add a translation to your native language, if that is not English? Also, is $f'$ the derivative of $f$? – dfeuer Dec 09 '13 at 06:14
  • Yes,$f'$ is derivative of $f$,and I have edit.Thank you – math110 Dec 09 '13 at 06:20
  • Yes,Thank you,That's my mean – math110 Dec 09 '13 at 06:36
  • One more question: you write this out starting with $f$, but it looks like actually you're trying to start with $h$, and find out if $f$ exists. Or did you mean to assume that $h$ is Frobenius, instead of assuming that $f$ is? – dfeuer Dec 09 '13 at 06:38
  • I rewrote the question. Please check if I did it right. If I made it all wrong, you can go to the edit history and revert. – dfeuer Dec 09 '13 at 06:52

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