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I need to decide if certain cyclotomic polynomials are irreducibles over the $\mathbb{F}_q$. For example, if $\Phi_{12}(x)$ is irreducible over $\mathbb{F}_9$. Anyone can help me?

Ok, i think i should aclare something: this question is not a duplicate of the question i made before... if you take the time to read my other question you will see that this is in fact a question that come up from that, because there was mention a criterion to decide if a cyclotomic polynomial was irreducible over $\mathbb{F}_q$, and i get an answer, but i note that the criterion requieres that $(n,q)=1$ so i made this question...please someone could give an idea?

Dimitri
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  • Please don't re-ask your own questions. – JSchlather Feb 16 '13 at 20:14
  • Sorry, but is not a duplicate, here i am asking for a case when $n$ and $q$ are not coprime. The other question was about a know criterion to decide irreducibility when they are in fact corpimes – Dimitri Feb 16 '13 at 21:21
  • Can anyone tell how to cange the "duplicate" of the question, becuase i feel that nobody is gonna ask this question anymore if they think is a duplicate...different question and a i need help – Dimitri Feb 16 '13 at 21:29
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    Dimitri, I'm not sure I follow in your first question Jyrki gave a criterion which said precisely when the cyclotomic polynomials are irreducible over $\mathbb F_q$. I don't see anywhere in his post where he assumes that $n$ and $q$ are coprime. – JSchlather Feb 16 '13 at 22:01
  • In the existence of a $k$ such that $n|q^{k}-1$, the fact that $(n,q)=1$ is required. As in this case $n=12$ and $q=9$ there is no such $k$ so you can´t apply that criterion. I made that aclaration in his answer, but nobody answer, so i made this question to see if there is a way to do this in this case – Dimitri Feb 16 '13 at 22:28

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The polynomial $\Phi_{12}=X^4-X^2+1$ is not even irreducible over ${\mathbb F}_3$. Indeed, we have modulo $3$, $\Phi_{12}=X^4+2X^2+1=(X^2+1)^2$. Over ${\mathbb F}_9$, $X^2+1$ has a root and can thus be completely factored.

Ewan Delanoy
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  • Thanks for the answer, i dont know if this will work for the other cases i should prove, but at least give me a better perspective – Dimitri Feb 17 '13 at 02:48