Show that {1, y, , y p- } is a basis for Z p (y) over Z p (y p ), where y is

Show that {1, y,· · ·, y^p-¹} is a basis for Z_p(y) over Z_p(y^p), where y is an indeterminate. Referring to Example 51.4, conclude by a degree argument that x^p -t is irreducible over Z_p(t), where t = y^p.

Data from in Example 51.4

Let E = Z_p(y), where y is an indeterminate. Let t = y^P, and let F be the subfield Z_p(t) of
E. (See Fig. 51.5.) Now E = F(y) is algebraic over F, for y is a zero of (x^p - t) ∈ F[x]. By Theorem 29.13, irr(y, F) must divide x^P -t in F[x]. [Actually, irr(y, F) = x^p — t. We leave a proof of this to the exercises (see Exercise 10).] Since F(y) is not equal to F, we must have the degree of irr(y, F) ≥ 2. But note that x^P - t = x^p - y^p = (x - y)^p,
since E has characteristic p. Thus y is
a zero of irr(y, F) of multiplicity > 1. Actually, x^P — t = irr(y, F), so the multiplicity of y is p.