Exercise 24, § 5

Let $k$ be a field, $A$ a $k$-algebra, $k'$ a radicial extension of $k$, $A' = A \otimes_k k'$. Now prove that the canonical mapping $\mathrm{Spec}(A') \to \mathrm{Spec}(A)$ is a homeomorphism. (When $A$ is a field, $A'$ is of dimension 0 (Exercise 20); we shall show that it possesses only one prime ideal. It will be deduced that for every prime ideal $p$ of $A$, the root $p'$ of $pA'$ is a prime ideal and that the mapping $p \mapsto p'$ is continuous.)