a galois related problem
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From: Derek Holt on 17 Feb 2010 11:46 On 17 Feb, 16:17, Kent Holing <K...(a)statoil.com> wrote: > Explicit example: > An irreducible monic equation with integer coefficients > of degree n, Galois group with 2n elements but not dihedral and at least 4 real roots (and both real and complex roots). This means n > 6. x^8 - 4*x^4 + 1 has 4 real roots, and its Galois group is isomorphism to D4 X C2 and is generated by the permutations: (1, 2)(3, 4)(5, 6)(7, 8) (1, 4)(2, 3)(5, 8)(6, 7) (2, 7)(3, 6) Derek Holt.
From: groupoide on 18 Feb 2010 02:23 On 12 fév, 12:11, Kent Holing <K...(a)statoil.com> wrote: > Can something general be said about the following: > For an irreducible equation with rational coefficents of degree n and both real and complex roots, what is the maximum number of real roots if the Galois group is Dn (2n elements)? > > Note that if r is a real root of such an equation then all other >real roots of the equation are in Q[r]. I do not understand. Why? Thanks
From: Kent Holing on 17 Feb 2010 16:40 Do you have an example with n is odd?
From: Derek Holt on 18 Feb 2010 03:48 On 18 Feb, 07:40, Kent Holing <K...(a)statoil.com> wrote: > Do you have an example with n is odd? Look, you must include some context with your post, either by quoting a previous post in the thread or by restating the problem. Derek Holt.
From: secondmouse on 18 Feb 2010 06:47
On Feb 18, 7:40 am, Kent Holing <K...(a)statoil.com> wrote: > Do you have an example with n is odd? P(x) = x^15 - 7*x^14 + 19*x^13 - 16*x^12 - 15*x^11 + 10*x^10 + 60*x^9 - 102*x^8 + 81*x^7 - 42*x^6 + 12*x^5 + 11*x^4 - 21*x^3 + 14*x^2 - 5*x + 1 which has signature [5,5] but the Galois group is not D15. This is taken from (and I've posted this before): http://www.math.uni-duesseldorf.de/~klueners/minimum/node562.html |