I never studied anything about these types of problems before so this is just a guess.

Say that,

....

.

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 .

For example,

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 .