Lemma 10.42.3. Let $k \subset K$ be a separably generated, and finitely generated field extension. Set $r = \text{trdeg}_ k(K)$. Then there exist elements $x_1, \ldots , x_{r + 1}$ of $K$ such that

1. $x_1, \ldots , x_ r$ is a transcendence basis of $K$ over $k$,

2. $K = k(x_1, \ldots , x_{r + 1})$, and

3. $x_{r + 1}$ is separable over $k(x_1, \ldots , x_ r)$.

Proof. Combine the definition with Fields, Lemma 9.19.1. $\square$

There are also:

• 4 comment(s) on Section 10.42: Separable extensions

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).