Lemma 9.26.5 (030H)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 9: Fields
Section 9.26: Transcendence
Lemma 9.26.5 (cite)

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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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