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Wikipedia / Algebraic independence

In abstract algebra, a subset of a field is algebraically independent over a subfield do not satisfy any non-trivial polynomial.

if and only if is transcendental, and over all of the field extensions.

Example

The real numbers 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, because the nontrivial polynomial

.

Algebraic independence of known constants

Although and e. is irrational. Nesterenko proved in 1996 that:

Results and open problems

The Lindemann–Weierstrass theorem are algebraic numbers that are linearly independent.

The Schanuel conjecture would establish the algebraic independence of many numbers, including and , but remains unproven:

  1. complex numbers that are linearly independent. The field extension has transcendence degree.

Algebraic matroids

Given a field extension that is not algebraic, Zorn's lemma. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.

For every finite set satisfy the axioms that define the independent sets of a matroid. 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, 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.

See also

References