Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In abstract algebra, a subset
is algebraically independent over a subfield
do not satisfy any non-trivial polynomial equation with coefficients in
In general, all the elements of an algebraically independent set
are by necessity transcendental over
generated by the remaining elements of
are transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers.
are both algebraically independent over the rational numbers.
is not algebraically independent over the rational numbers
, because the nontrivial polynomial is zero when
Although π and e are transcendental, it is not known whether
[2] Nesterenko proved in 1996 that: The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over
are algebraic numbers that are linearly independent over
The Schanuel conjecture would establish the algebraic independence of many numbers, including π and e, but remains unproven: Given a field extension
that is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of
Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
For every finite set
, the algebraically independent subsets of
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
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.
[5] Many finite matroids may be represented by a matrix over a field
, 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.