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

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

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

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

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