Given a field extension L/K, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of L over K. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
For every finite set S of elements of L, the algebraically independent subsets of S satisfy the axioms that define the independent sets of a matroid.
In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set T of elements is the intersection of L with the field K[T].
[4][5] Many finite matroids may be represented by a matrix over a field K, in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent.
Every matroid with a linear representation of this type over a field F may also be represented as an algebraic matroid over F,[6][7] by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals.