A polynomial whose galois group is D_8 - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T21:35:34Z http://mathoverflow.net/feeds/question/105811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105811/a-polynomial-whose-galois-group-is-d-8 A polynomial whose galois group is D_8 Dharam 2012-08-29T09:54:36Z 2012-09-10T10:21:32Z <p>I need to construct such a polynomial, and more generally: given a group G, how can it be realized as a galois group?</p> http://mathoverflow.net/questions/105811/a-polynomial-whose-galois-group-is-d-8/105855#105855 Answer by quid for A polynomial whose galois group is D_8 quid 2012-08-29T15:57:28Z 2012-08-29T15:57:28Z <p>For the case that <code>\$D_8\$</code> 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.</p> <p>For the other interpretation of <code>\$D_8\$</code> a polynomial with minimal discriminat would be \$\$x^4 - x^3 -x^2 +x +1 \$\$</p> <p>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). </p> <p>The above polynomials and information are taken from the <a href="http://galoisdb.math.upb.de/home" rel="nofollow">database of Jürgen Klüners and Gunter Malle</a> 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).</p> http://mathoverflow.net/questions/105811/a-polynomial-whose-galois-group-is-d-8/106776#106776 Answer by Gene Ward Smith for A polynomial whose galois group is D_8 Gene Ward Smith 2012-09-10T04:34:07Z 2012-09-10T04:34:07Z <p>For Galois groups up through degree eight, you can easily provide an infinte number of examples by the function field extensions listed here:</p> <p><a href="http://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01160-6/S0025-5718-99-01160-6.pdf" rel="nofollow">Some polynomials over Q(t) and their Galois groups</a></p> http://mathoverflow.net/questions/105811/a-polynomial-whose-galois-group-is-d-8/106800#106800 Answer by Inta for A polynomial whose galois group is D_8 Inta 2012-09-10T10:21:32Z 2012-09-10T10:21:32Z <p>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)</p>