Algebraic independence

In abstract algebra, a subset formula_1 of a field formula_2 is algebraically independent over a subfield formula_3 if the elements of formula_1 do not satisfy any non-trivial polynomial equation with coefficients in formula_3.
In particular, a one element set formula_6 is algebraically independent over formula_3 if and only if formula_8 is transcendental over formula_3. In general, all the elements of an algebraically independent set formula_1 over formula_3 are by necessity transcendental over formula_3, and over all of the field extensions over formula_3 generated by the remaining elements of formula_1.
Example.
The two real numbers formula_15 and formula_16 are each transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, each of the two singleton sets formula_17 and formula_18 are algebraically independent over the field formula_19 of rational numbers.
However, the set formula_20 is "not" algebraically independent over the rational numbers, because the nontrivial polynomial
is zero when formula_22 and formula_23.
Algebraic independence of known constants.
Although both formula_24 and "e" are known to be transcendental,
it is not known whether the set of both of them is algebraically independent over formula_19. In fact, it is not even known if formula_26 is irrational.
Nesterenko proved in 1996 that:
Lindemann–Weierstrass theorem.
The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over formula_19. It states that whenever formula_38 are algebraic numbers that are linearly independent over formula_19, then formula_40 are also algebraically independent over formula_19.
Algebraic matroids.
Given a field extension formula_42 which is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of formula_2 over formula_3. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
For every set formula_1 of elements of formula_2, the algebraically independent subsets of formula_1 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 formula_48 of elements is the intersection of formula_2 with the field formula_50. A matroid that can be generated in this way is called an algebraic matroid. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid.
Many finite matroids may be represented by a matrix over a field formula_3, 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 may also be represented as an algebraic matroid, 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. The converse is false: not every algebraic matroid has a linear representation.