transcendence basis

A basis of a vector space is a maximal collection of independent directions. A transcendence basis is the exact analogue for fields, but with 'algebraically independent' replacing 'linearly independent'. It is a set of elements that satisfy no polynomial relation among themselves over the base field, and that is as large as such a set can be: throw in any further element and a polynomial relation appears. It records the genuinely free 'variables' inside an extension, after which everything else is algebraic.

Precisely, given an extension L of K, a transcendence basis is a subset S of L that is algebraically independent over K and such that L is algebraic over K(S). Equivalently, S is a maximal algebraically independent subset. Algebraic independence means no nonzero polynomial in finitely many of the elements of S, with coefficients in K, evaluates to zero.

Every field extension has a transcendence basis, by a Zorn's-lemma argument exactly parallel to the existence of vector-space bases, and any two transcendence bases have the same cardinality — this common size is the transcendence degree. The extension then factors as a purely transcendental part K(S) followed by an algebraic part L over K(S), which is the structural reason the notion is so useful.