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I'm trying to find the orbits of the Frobenius homomorphism $\phi_p: a \to a^p$ on $\mathbb{F}_{p^4} / \mathbb{F}_p$. I can see there are $p$ 1 orbits corresponding to the action on $\mathbb{F}_p$ but I think there are also some order 2 and 4 orbits but I'm having trouble finding how many of each there are.

This is part of an argument on trying to show how many irreducible polynomials of degree 4 there are over $\mathbb{F}_p$ because I think this number should correspond to how many order 4 orbits there are, as then the polynomial with roots in that orbit should be irreducible.

Thanks

Wooster
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  • Do you speak german? Page 16 (in actual page numbers of the paper: 12) of this paper will interest you: http://page.math.tu-berlin.de/~kant/Algebra/algalg3.pdf – kummerer94 Jan 06 '15 at 21:57
  • I'm afraid my german isn't quite up to it, but thank you for the reference – Wooster Jan 06 '15 at 22:05
  • And I'm afraid it isn't quite what you wanted but it shows a way to find all irreducible polynomials in $\mathbb{F}_p$ - at least for degree $\leq 4$.

    The main theorem is: In $ \mathbb{F}_q$ where $q = p^r$ for $p$ prime and $r\gt 0$, $t^{q^m} - t $ is the product of all normalized irreducible polynomials $ f \in \mathbb{F}_q [t]$ where $\deg f | m $.

     – kummerer94 Jan 06 '15 at 22:06
  • http://math.stackexchange.com/questions/40811/number-of-monic-irreducible-polynomials-of-degree-p-over-finite-fields isn't the same question, but one of the answers answers your question. – Gerry Myerson Jan 06 '15 at 22:17
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    A comment at http://math.stackexchange.com/questions/152880/how-many-irreducible-polynomials-of-degree-n-exist-over-mathbbf-p indicates the question has been asked here many times. – Gerry Myerson Jan 06 '15 at 22:18
  • I have seen those posts but I'm looking more to prove this specific result using the method I've started to outline rather than the general result if possible. I.e perhaps the question would be better phrased as how many order 2 orbits in $\mathbb{F}_p$ under $\phi_p$ are there – Wooster Jan 06 '15 at 22:47
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    I took back my vote to close. But, yes, you should really make/have made clear that you are mainly interested in making your approach work and not in the result itself giving this as your attempt. – quid Jan 06 '15 at 23:40
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    Edit your question to make the non-duplication crystal clear, then go to the meta site and find the thread where we make requests for reopening questions, and make the case there for reopening. And next time, be a little more forthcoming when you post a question, please. – Gerry Myerson Jan 07 '15 at 03:35
  • I've now edited the question, I can't find this thread you mention, would you mind providing a link? – Wooster Jan 07 '15 at 08:52
  • http://meta.math.stackexchange.com/questions/19042/requests-for-reopen-undeletion-votes-etc-new-version (for future reference, as I see the question has already been re-oopened). – Gerry Myerson Jan 07 '15 at 16:38

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