Exercise 33, § 2

Let K be a commutative field (of arbitrary characteristic q), and let $l$ be a prime number. Let K' be an algebraic extension of K with the following two properties: (i) every polynomial in $K[X]$ whose degree is not divisible by $l$ has a root in K'; (ii) every polynomial in $K'[X]$ of degree equal to $l$ has a root in K'. a) Show that the perfect closure $K_1$ of K is contained in K' and satisfies the condition analogous to (i). b) Every algebraic extension of K whose degree is not divisible by $l$ is isomorphic to a subextension of K' (use the primitive element theorem). c) Let L be a subfield of K' containing K, and let M be a Galois extension of L, whose degree is a power of $l$. Show that M is isomorphic to a sub-extension of K' (use the fact that an I-group is nilpotent). d) Deduce that K' is an algebraic closure of K (if F is a (separable) extension of K of finite degree, consider a Galois extension F, $\supset F$ of finite degree, and the fixed subfield of F, under a Sylow I-subgroup of the Galois group of F, over K).