Determine the group of automorphisms of Q2SolutionConsider t

Determine the group of automorphisms of Q(2)

Solution

Consider the ring Q[2–]Q2. It is a vector space over QQ and as such any automorphism, , of Q[2–]Q2 is determined by its basis elements, in this case {1,2–}12. Since this is a ring homomorphism we have 1111 and the automorphism is completely determined by the image of 2–2.

Furthermore, we must have (2–)2=2222 since all elements of QQ are invariant under due to the image of 11. Denoting x=(2–)x2 we have x22=0x220 and x=±2–x2. This gives the complete set of automorphisms of Q[2–]Q2, namely the two maps 2–2–22 and 2–2–22.