Exercise 2, § 7

Let k be an imperfect field of characteristic p, a \in k an element such that a \notin k^p, F the field k(X) of rational fractions in one indeterminate over k; put y = X^{p^2}/(X^p + a) and K = k(y).

a) Show that [F:K] = p^2, so that T^{p^2} - yT^p - ay is the minimal polynomial of X in K[T]; deduce that the relative separable closure of K in F is E = k(X^p) (cf. V, p. 148, Ex. 11).

b) Show that F is an exceptional extension (Ex. 1) of K. (Argue by contradiction, assuming b \in F such that b \notin K and b^p \in K. Observe that K(b) = k(z), where z = r(X)/s(X) with deg r = p, deg s \leq p and r, s being relatively prime polynomials in k[X]; show that z^p \in K and that the minimal polynomial of X over K is r(T)^p - z^p s(T)^p up to a factor from K. Conclude, using a) that we would have a \in k \cap F^p = k^p contrary to the hypothesis.)