Originally Posted by

**dangkhoa** I've been asked to decide whether the following extensions are normal

(a) Q(cube root of 5,i):Q

(b) Q(5^(1/4),i):Q

(c) C(t):C(t^3)

(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