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What is the best method to find the Galois group of $P = X^4+4X^3+12X^2+24X+24$ over $\mathbb{Q}$ ?

First, I don't manage to show that $P$ is irreducible : Eisenstein doesn't work.

I know that its discriminant is a square, so, if we assume that $P$ is irreducible, $Gal(P, \mathbb{Q}) \subset A_4$. But if we reduce $P$ modulo $2$ or $3$, we don't obtain something interesting (I think).

Arnaud
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    To show it is irreducible (over $\Bbb Q$) prove it doesn't have any rational roots and that it can't be the product of two polynomials of degree $2$ by solving a system. – Git Gud Apr 07 '13 at 12:17
  • Really, solving a system ? There is no a quicker method ? – Arnaud Apr 07 '13 at 12:29
  • I forgot to mention that it's better if you use the above with the second Gauss's Lemma. Also remember you can suppose the two polynomials are monic. If you want you can even go $\pmod p$. – Git Gud Apr 07 '13 at 12:30
  • There might be some linear substitution that works, but the simpler ones don't work and others I wouldn't really classify them as 'faster'. – Git Gud Apr 07 '13 at 12:32
  • Ok thank you. So, assume that I've done it. How to compute the Galois group now ? – Arnaud Apr 07 '13 at 12:35
  • Do you know the method by Dedekind-Frobenius? Also consult http://math.stackexchange.com/questions/220310/tips-for-finding-the-galois-group-of-a-given-polynomial – Martin Brandenburg Apr 07 '13 at 13:03
  • No, I don't know it. I just follow an introductory Galois Theory course. – Arnaud Apr 08 '13 at 06:55

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