What is the Galois group of a polynomial over a finite field?

Index of Splitting Field (galois Theory)?

• Suppose that the polynomial ax^4 + bx^2 + c for some a,b,c in Q (rationals) is irreducible over Q and let K be a splitting field over Q for it. Prove that [K:Q] is either 4 or 8. I'm trying to work through the different cases of the roots of the polynomial (i.e. 4 distinct roots, 1 repeated root etc) but I'm not really getting very far.... Can anyone help? :)

• Answer:

  When you adjoin one root r then you get an extension of degree 4. Now either this is the splitting field or it isn't. If it isn't, then the polynomial now factors as ax^4 + bx^2 + c = (x - r)(x + r)f(x), where f(x) is a quadratic irreducible over your new field. So adjoin a root for f(x) and you're done; your last field will be degree two over the first extension, so it will be degree 8 over Q.