Recent content by dave_hopkins

Thanks! I am looking for a polynomial x^4+ax^2+b, that has a Galois group that is the dihedral group of order 8. Thus, I imagine, it must have roots a1,a2,a3,a4, where a1+a2 = sqrt{j}, a1 + a3 = sqrt{k} and a1 + a4 = sqrt{l}, with sqrt{j}, sqrt{k}, sqrt{l} irrational. pretty stuck. any help...

yeh, you have eisenstein's criterion correct.

Thanks. I'm just trying to work out how you found this out. x^4-10x^2+1 is factorized as (x^2 + a)(x^2 + b), where a = -5 +/- 4\sqrt{6} and b = -5 -/+\sqrt{6}. The four roots (a1,a2,a3,a4) are then the +/- roots of a. And q = a1+a2 w=a1 + a3, e = a1 +a4 make up the fixed field Q(q^2,w^2,e^2)...

This does do it. We were considering polynomials which are irreducible in Q, but this is the Galois group of the polynomial f = ((x^2)-2)((x^2)-3), which is clearly not irreducible. However, you are correct. I am looking for a polynomial (preferably quintic) to research that is irreducible and...