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 .
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 are 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 . In fact, it is not even known if is irrational. Nesterenko proved in 1996 that:
- the numbers , , and Γ(1/4) are algebraically independent over .
- the numbers , , and Γ(1/3) are algebraically independent over .
- for all positive integers , the numbers and are algebraically independent over .
The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over . It states that whenever are algebraic numbers that are linearly independent over , then are also algebraically independent over .
Given a field extension which 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.
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.
- Patrick Morandi (1996). Field and Galois Theory. Springer. p. 174. ISBN 978-0-387-94753-2. Retrieved 2008-04-11.
- Green, Ben (2008), "III.41 Irrational and Transcendental Numbers", in Gowers, Timothy (ed.), The Princeton Companion to Mathematics, Princeton University Press, p. 222
- Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. 49 (Second ed.). p. 61. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
- Nesterenko, Yuri V (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914.
- Ingleton, A. W.; Main, R. A. (1975), "Non-algebraic matroids exist", Bulletin of the London Mathematical Society, 7: 144–146, doi:10.1112/blms/7.2.144, MR 0369110.
- Joshi, K. D. (1997), Applied Discrete Structures, New Age International, p. 909, ISBN 9788122408263.