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

closed as too localized by Steven Landsburg, Todd Trimble♦, Alex Bartel, JSE, Felipe Voloch Sep 10 '12 at 23:13
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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). 


Over $\Bbb Q$, F. Seidelmann, in "Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigen Rationalitätsbereich, Math. Ann. 78, 230233 (1917)", gives the following parametric representation of degree $4$ equations with group $D_8$: $$x^42(e^2f+g)x^24efx+[(e^2fg)^2f]=0$$ (with some restrictions on the parameters) 


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

