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## Main Question or Discussion Point

Let K = Q(2^(1/4))

a) Which of the morphisms from K to C are Q(2^1/2)-homomorphisms

b) And which are K-homomorphisms?

Attempt at a solution

Ok, I don't really understand this very well but for a) I know that there are 4 homomorphisms, since the minimal polynomial over C has four solutions and there is a bijection between the roots and the homomorphisms. What I don't understand is how I get from the number of homomorphisms to the homomorphisms themselves. If someone could explain that to me I think it would really help.

b) I can't really do b) until I know how to get the homomorphisms

I do not want to push my luck as I would be really happy if someone could give me some pointers on the previous questions, but if there was someone who didn't mind helping out a struggling student any pointers on the following would be greatly appreciated also.

c) Determine the automorphism group Aut(K/Q)

d) Find an element in K that is not in Q and that is fixed by every element of Aut(K/Q)

e) Conclude that K/Q is not Galois

a) Which of the morphisms from K to C are Q(2^1/2)-homomorphisms

b) And which are K-homomorphisms?

Attempt at a solution

Ok, I don't really understand this very well but for a) I know that there are 4 homomorphisms, since the minimal polynomial over C has four solutions and there is a bijection between the roots and the homomorphisms. What I don't understand is how I get from the number of homomorphisms to the homomorphisms themselves. If someone could explain that to me I think it would really help.

b) I can't really do b) until I know how to get the homomorphisms

I do not want to push my luck as I would be really happy if someone could give me some pointers on the previous questions, but if there was someone who didn't mind helping out a struggling student any pointers on the following would be greatly appreciated also.

c) Determine the automorphism group Aut(K/Q)

d) Find an element in K that is not in Q and that is fixed by every element of Aut(K/Q)

e) Conclude that K/Q is not Galois