It must have this form because, it cannot be that degree of t exceedes degree of s for that would make the polynomial of degree s exceedes degree of t which means it cannot be the same after applying the automorphism to it.
Now this value stays unchanged under . The values across the diagnols can be anything while .
We can pair it as,
This seems to suggest that,
But it turns out that .
Because can be obtained from for . For example, . And . And so on.
Thus. the fixed field under the group is .