Bug in any_root introduced in #37170

[grhkm21]

Steps To Reproduce

The trace computation in the helper method _cantor_zassenhaus_split_to_irreducible introduced by #37170 is implemented incorrectly. More specifically, these lines at https://github.com/sagemath/sage/blob/develop/src/sage/rings/polynomial/polynomial_element.pyx#L2176-L2177

            if q % 2:
                h = self.gcd(pow(T, (q-1)//2, self) - 1)

should be

            if q % 2:
                h = self.gcd(pow(T, (q**degree-1)//2, self) - 1)

Proof:

sage: R.<x> = GF(13)[]
....: def irr(d, R): return f.monic() if (f := R.random_element(d)).is_irreducible() else irr(d, R)
....: f = prod(irr(6, R) for _ in range(10))
....: f.factor()[0][0].degree()
....: f.any_root(degree=6) # errors

Expected Behavior

don't error

Actual Behavior

error

Additional Information

We have a fix already

Environment

Latest `develop` branch

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide