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Lemma 10.42.3. Let $K/k$ 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. $\square$

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