transcendence degree

Noun

 * 1)  Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.
 * 2) * 2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, 3076, page 263,
 * Let $$F_1/k$$ and $$F_2/k$$ denote algebraic function fields of transcendence degree one.

Usage notes

 * A transcendence degree is said to be of a field extension (i.e., $$L/K$$). More properly, it is the cardinality of a particular type of subset of the extension field $$L$$, although the context of the field extension is required to make sense of the definition.
 * Relatedly, a of $$L/K$$ is a subset of $$L$$ that is algebraically independent over $$K$$ and such that $$L$$ is an algebraic extension of $$K(S)$$ (that is, $$L/K(S)$$ is an algebraic extension).
 * It can be shown that every field extension $$L/K$$ has a transcendence basis, whose cardinality, denoted $$\operatorname{trdeg}_K L$$ or $$\operatorname{trdeg}(L/K)$$, is exactly the transcendence degree of $$L/K$$.