In particular, a one element set is algebraically independent over if and only if is
transcendental over . In general, all the elements of an algebraically independent set over are by necessity transcendental over , and over all of the
field extensions over generated by the remaining elements of .
Example
The two
real numbers and 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 and is algebraically independent over the field of rational numbers.
However, the set is not algebraically independent over the rational numbers, because the nontrivial polynomial
is zero when and .
Algebraic independence of known constants
Although both and
e are known to be transcendental,
it is not known whether the set of both of them is algebraically independent over .[1] In fact, it is not even known if is irrational.[2]Nesterenko proved in 1996 that:
the numbers , , and , where is the
gamma function, are algebraically independent over .[3]
the numbers and are algebraically independent over .
for all positive integers , the number is algebraically independent over .[4]
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 over . Further, all the maximal algebraically independent subsets have the same
cardinality, known as the
transcendence degree of the extension.
For every set of elements of , 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 of elements is the intersection of with the field . 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.[6]