I've been asked to decide whether the following extensions are normal
(a) Q(cube root of 5,i):Q
(d) Q(t): Q(t^3)
This is my attempts
(a) Let a = 5^(1/3), then a has minimal polynomial m(x) = x^3 -5 which is irreducible over Q by Eisenstein's criterion with p = 5
i has min poly x^2 +1 over Q(5^(1/3)) and since i does not belong to Q(5^(1/3)) , then it is irreducible over Q(5^(1/3))
Then by tower law, [Q(cube root of 5,i):Q] = 3*2 = 6
Also, the polynomial f(x) = x^3 -5 = (x -a)(x-aw)(x-a(w^2)) where w is the cube root of unity.
Is it correct so far?
I've done up to here and dont know how to decide whether normal or not.
Part b is similar to part a
But bart c and d looks a bit hard. Can you give me some hints on these parts please?
Thank you very much