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If I have a polynomial which I've factorised into irreducibles over GF(p), p prime, and it doesn't have any repeated factors, then what is its Galois group over this finite field (and what is the cycle type of its generator)?

for example, what is the galois group of $(X^3 + 4)(X^2+2)(X+1)$ over GF(7)?

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closed as too localized by S. Carnahan, Reid Barton, Mariano Suárez-Alvarez, Qiaochu Yuan, Gjergji Zaimi Apr 15 '10 at 21:40

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I think that this is elementary algebra and does not belong to MO. please check out the faq (mathoverflow.net/faq). –  Martin Brandenburg Apr 15 '10 at 14:52
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Indeed -- this sort of thing is well-suited to the appropriate forum at Art of Problem Solving: in this case, that's artofproblemsolving.com/Forum/index.php?f=8 –  JBL Apr 15 '10 at 15:09
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Voting to close. Finite extensions of finite fields only admit cyclic galois groups. See en.wikipedia.org/wiki/Finite_field –  S. Carnahan Apr 15 '10 at 15:19
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@Scott: that doesn't contradict anything in the question. What jsmith might mean is what is the automorphism group of $\mathbb{F}_7 [x]/(f)$, and that need not be a cyclic group. If he means the automorphism group of the splitting field, then while the answer is cyclic, it's size is related to the factors and not just the total degree. –  Dror Speiser Apr 15 '10 at 16:30
    
I suppose one could make that interpretation, but I think jsmith12 should have given a precise definition in the statement of the question. Assuming the more generous interpretation of "Galois group", the automorphism group of an etale algebra over a finite field is a product of wreath products of cyclic groups by symmetric groups. –  S. Carnahan Apr 15 '10 at 23:51