Describe the value of the Frobenius automorphism 3 on each element of the finite field of

Describe the value of the Frobenius automorphism σ₃ on each element of the finite field of nine elements given in Exercise 18 of Section 29. Find the fixed field of σ₃.

Data from Exercise 18 Section 29

Show that the polynomial x² + 1 is irreducible in Z₃[x]. 

Let a be a zero of x² + 1 in an extension field of Z₃. As in Example 29.19, give the multiplication and addition tables for the nine elements of Z₃(α), written in the order 0, 1, 2, α, 2α, 1 + α, 1 + 2α, 2 + α, and 2 + 2α.