nLab transcendence degree

Definition

Let $k$, $E$ be fields, with $k \hookrightarrow E$ a field extension, also written as $E/k$. The transcendence degree of $E/k$ is the cardinality of a maximal set of elements of $E$ that are algebraically independent over $k$.

The transcendence degree is well-defined, i.e., independent of which maximal set of algebraically independent elements is used. This is often proven by invoking a Mac Lane-Steinitz exchange condition; see matroid for a general argument.

Examples

Given a countable field $K$ and any infinite set $S$, the transcendence degree of the field extension $K(S)$ will be the same as the cardinality of the generating set $S$, and any algebraic extension of $K(S)$ will have the same cardinality again.)

Related concepts

• field extension