- #1

wattsup03

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## Homework Statement

Let K = Q(2

^{1/4})

Determine the automorphism group Aut(K/Q)

## Homework Equations

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.

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