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I need to construct such a polynomial, and more generally: given a group G, how can it be realized as a galois group?

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As Konstantin says, this problem is very difficult in general. But it is doable for certain subgroups of $S_n$ for low $n$. Do you know how to compute Galois groups of cubics and quartics? As for your example, I think $x^4-2x^2-2$ works. –  J.C. Ottem Aug 29 '12 at 10:04
Not over the complex numbers. Dharam: If you write $D_8$, do you mean the dihedral group with 8, or with 16 elements? –  René Aug 29 '12 at 10:29
This question has also been asked in math.stackexchange.com/questions/188296/… –  lhf Aug 29 '12 at 11:09
Personally, I believe that it should be acceptable to ask "basic" questions on mathoverflow rather than math.SE if the asker prefers more concise answers using higher-powered machinery, or if the question has been asked on math.SE without receiving a good answer. I personally would probably even consider acceptable a question that was asked simultaneously on both sites, if the asker linked the two questions and gave a good explanation for why they were posting simultaneously. However, since this question was cross-posted simultaneously with no explanation, I am voting to close. –  Charles Staats Aug 29 '12 at 16:12
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closed as too localized by Steven Landsburg, Todd Trimble, Alex Bartel, JSE, Felipe Voloch Sep 10 '12 at 23:13

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3 Answers

For the case that $D_8$ is interpreted to be with $16$ elements (which is the case according to OP on M.SE) an example is given by $$ x^8 - 3 x^5 - x^4 + 3x^3 +1 $$ This is a polynomial with minimal (in absolute value) discriminant with that Galois group.

For the other interpretation of $D_8$ a polynomial with minimal discriminat would be $$x^4 - x^3 -x^2 +x +1 $$

As said in the comments the general problem is difficult (indeed open even regarding existence, that is over the rationals, which I assume is the intention of the question).

The above polynomials and information are taken from the database of Jürgen Klüners and Gunter Malle where a great many examples and information can be found, presented in a nice way. So, for examples for specific not too large groups one migt well find a polynomial there, even if one wishes additional restrictions (on the siganture, say).

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I didn't know of the database. Thank you. –  Andres Caicedo Aug 29 '12 at 16:19
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Over $\Bbb Q$, F. Seidelmann, in "Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigen Rationalitätsbereich, Math. Ann. 78, 230--233 (1917)", gives the following parametric representation of degree $4$ equations with group $D_8$: $$x^4-2(e^2f+g)x^2-4efx+[(e^2f-g)^2-f]=0$$ (with some restrictions on the parameters)

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For Galois groups up through degree eight, you can easily provide an infinte number of examples by the function field extensions listed here:

Some polynomials over Q(t) and their Galois groups

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I had assumed OP meant over $\mathbb{Q}$, although I grant you that wasn't specified in the question. –  Todd Trimble Sep 10 '12 at 14:43
The point is, if you have F/Q(t) with Galois group G, specializing t to a value in Q leads to an extension K/Q with Galois group G in nearly all cases. Hence it's a sort of machine for cranking out endless examples. It is particularly nice if F contains no algebraic extension of Q, as then the examples your machine cranks out don't share a common extension of Q. –  Gene Ward Smith Sep 10 '12 at 21:25
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