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 $k$. $\square$

There are also:

• 9 comment(s) on Section 9.26: Transcendence

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).