Exercise 11, § 14

Let E be an extension of a field K and $\Omega$ an algebraically closed extension of E. Show that the following conditions are equivalent : a) E is a p-radical extension of K ; β) for every extension F of K, $E \otimes_K F$ has only a single prime ideal ; γ) for every extension F of K, any two composite extensions of E and F are isomorphic ; δ) $E \otimes_K \Omega$ has only a single prime ideal ; ε) any two composite extensions of E and $\Omega$ are isomorphic.