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(1) id is the identity

(2) tau maps sqrt2 to -sqrt2 and leave 3i alone

(3) sigma leaves sqrt2 alone and maps 3i to -3i

(4) gamma maps sqrt2 to -sqrt2 and 3i to -3i ?

If so, the what are the fixed fields of the subgroups?

If I'm not mistaken the (proper nontrivial) subgroups are H={id, tau}, J={id, sigma}, K={id, gamma}. It appears that the fixed field of H is Q(3i) and the fixed field of J is Q(sqrt2). But it also appears that the fixed field of K is just Q, which is also the fixed field of Gal(F/Q). But F/Q is Galois since F is the splitting field of a seperable polynomial, so we can't have two distinct groups associated to the same intermediate field.

What am I doing wrong?