Exercise 20, § 5

Let $k$ be a field, $k_i$ ($i = 1, 2$) two extensions of $k$, $n_i$ ($i = 1, 2$) the degree of transcendence of $k_i$ over $k$ if it is finite or the symbol $+ \infty$ otherwise. Now prove that one has

and that $k_1 \otimes_k k_2$ is noetherian if one of the extensions $k_i$ is of finite type.