Let K = Q(21/4)
Determine the automorphism group Aut(K/Q)
An automorphism is an isomorphism from a Field to itself
Aut(K/Q) is the group of Automorphisms from k/Q to K/Q
Definition: A K-Homomorphism from L/K to L'/K is a homomorphism L---> L' that is the identity on K
The Attempt at a Solution
I am completely at a loss really. I have calculated there are four homomorphisms from K to C and think from there if I know how many are K-homomorphisms then that'll be the number of automorphisms, because a homomorphism from a field to itself is an automorphism (Please correct me if I'm wrong on this). Then that'll give me the set of Automorphisms.
My problem is that I don't know how to go from the number of homomorphisms to the actual homomorphisms. I think it has a relation to the roots of 2(1 /4) in C (which I have calculated to be 2(1 /4), - 2(1 /4), i*2(1 /4), -i2(1 /4) )
Please help, this lack of understanding is preventing me from moving forward with other questions and my notes from lectures completely gloss over how to do this.