Algebraically independent

In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.

In particular, a one element set {α} is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S.

Example

The two real numbers \sqrt{\pi} and 2\pi+1 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 \{\sqrt{\pi}\} and \{2\pi+1\} are algebraically independent over the field \mathbb{Q} of rational numbers.

However, the set \{ \sqrt{\pi}, 2\pi+1 \} is not algebraically independent over the rational numbers, because the nontrivial polynomial

P(x,y)=2x^2-y+1

is zero when x=\sqrt{\pi} and y=2\pi+1.

Algebraic independence of known constants

Although both \pi and e are known to be transcendental, it is not known whether the set of both of them is algebraically independent over \mathbb{Q}.[1] In fact, it is not even known if \pi+e is irrational.[2] Nesterenko proved in 1996 that:

  • the numbers π, eπ, and Γ(1/4) are algebraically independent over Q.[3]
  • the numbers π, eπ√3, and Γ(1/3) are algebraically independent over Q.
  • for all positive integers n, the numbers π, eπ√n are algebraically independent over Q.[4]

Lindemann–Weierstrass theorem

The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over Q. It states that whenever α1,...,αn are algebraic numbers that are linearly independent over Q, then eα1,...,eαn are algebraically independent over Q.

Algebraic matroids

Main article: Algebraic matroid

Given a field extension L/K, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of L over K. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.

For every set S of elements of L, the algebraically independent subsets of S 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 T of elements is the intersection of L with the field K[T]. 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 K, 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]

References

External links

  • Chen, Johnny, "MathWorld.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.