Thread: Splitting fields

I am having trouble understanding this topic in algebra. Can you guys please help?

My question is what is the splitting field for x^3-2 over Q. In my book there is an example that says for x^n-a over Q a^1/n, wa^1/n, ..., w^n-1 * a^1/n are zeroes of x^n-a in Q(a^1/n, w) (w is primitive nth root of unity)


So does that mean that the splitting field for my problem would be: Q(2^1/3, w)? why?

yes, if ω is a primitive cube root of unity (a root of x^2 + x + 1 = (x^3 - 1)/(x - 1).).

Q(2^(1/3)) only contains the single real root of x^3 - 2. there are two other complex roots: (2^(1/3))ω, and (2^(1/3))ω^2.

clearly any splitting field K for x^3 - 2 has to contain these three roots, so Q(2^(1/3),ω) is contained in K.

on the other hand, Q(2^(1/3),ω) is the smallest extension of Q containing all three roots. this field is of degree 6 over Q.