Prove that . Conclude that has degree 4 over . Find an irreducible polynomial satisfied by .
Recall that has degree 2 over . Now we claim that has degree 2 over . To see this, suppose to the contrary that . Say for some rationals and . Coparing coefficients, we have and ; if , then , and if , then . Either case yields a contradiction.
So has degree 4 over .
The minimal polynomial of has degree 4 over . We can solve the system over , and in so doing we see that is the minimal polynomial of .