Lemma 9.26.5. Let $L/K/k$ be field extensions. Then

\[ \text{trdeg}_ k(L) = \text{trdeg}_ K(L) + \text{trdeg}_ k(K). \]

**Proof.**
Choose a transcendence basis $A \subset K$ of $K$ over $k$. Choose a transcendence basis $B \subset L$ of $L$ over $K$. Then it is straightforward to see that $A \cup B$ is a transcendence basis of $L$ over $k$.
$\square$

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