I was thinking about this, finding the splitting field of x^4-2 in Q[x] over Q is standard enough... but would much be different is i wanted the splitting field over F_5? (field with 5 elements) would it just be F_5(2^(1/4), i) analogously to the Q case? or do any of the arguments break down? Any thoughts are appreciated, cheers
I think it would be just F_5(2^(1/4)), since once you have one fourth root of 2, the others would just be 2*2^(1/4), 4*2^(1/4), and 3*2^(1/4) (since 2^4 = 1 in F_5).
Very true! However, what does 2^(1/4) mean exactly in this case? i dont think it can be a real number since i dont believe there is an extension from F_5 to R... And since there is no element x in F_5 such that x^4=2... perhaps i am confused? Thanks for you reply!
No, it wouldn't be an element of R. It would be an element of some algebraic extension of F_5. In this case, since the polynomial x^4 - 2 is irreducible over F_5, we can take that extension to be the quotient ring F_5[X]/(X^4 - 2).