Transcendence degree

Remarks

The definition makes sense because a transcendence basis exists and any two transcendence bases have the same cardinality; the proof uses an exchange argument like the Steinitz exchange lemma. Example: $\operatorname{trdeg}_K K(x_1, \dots, x_n) = n$, with transcendence basis $\set{x_1, \dots, x_n}$. Example: $\operatorname{trdeg}_\Q \R$ is the cardinality of the continuum, since $\R$ is uncountable and the algebraic closure of a countable field is countable. Non-example: for $L = \Q(\pi, \pi^2)$, the transcendence degree over $\Q$ is $1$, not $2$, because $\set{\pi, \pi^2}$ is dependent and $\set{\pi}$ is a transcendence basis. It is additive in towers: $\operatorname{trdeg}_K M = \operatorname{trdeg}_L M + \operatorname{trdeg}_K L$ for $K \subseteq L \subseteq M$. The transcendence degree is $0$ if and only if the extension is algebraic.