Algebraic independence

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". This means that for every finite sequence α₁, ..., α_"n" of elements of "S", no two the same, and every non-zero polynomial "P"("x"₁, ..., "x"_"n") with coefficients in "K", we have:"P"(α₁,...,α_"n") ≠ 0.

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 over "K" are by necessity transcendental over "K", but that is far from being a sufficient condition.

For example, the subset {√π, 2π+1} of the real numbers R is "not" algebraically independent over the rational numbers Q, since the non-zero polynomial

:$P(x\_1,x\_2)=2x^2\_1-x\_2+1$

yields zero when √π is substituted for "x"₁ and 2π+1 is substituted for "x"₂.

The Lindemann-Weierstrass theorem can often be used to prove that some sets are algebraically independent over Q. It states that whenever α₁,...,α_"n" are algebraic numbers that are linearly independent over Q, then "e"^α₁,...,"e"^α_"n" are algebraically independent over Q.

It is not known whether the set {π, e} is algebraically independent over Q. [cite book
url = http://books.google.ca/books?id=jQ7c8Xqpqk0C
title = Field and Galois Theory
author = Patrick Morandi
publisher = Springer
year = 1996
pages = 174
accessdate = 2008-04-11]
Nesterenko proved in 1996 that {π, "e"^π, Γ(1/4)} is algebraically independent over Q. [cite journal|author=Nesterenko, Yuri V|authorlink=Yuri Valentinovich Nesterenko|title=Modular Functions and Transcendence Problems|journal=Comptes rendus de l'Académie des sciences Série 1|volume=322|number=10|pages=909–914|year=1996]

Given a field extension "L"/"K", we can use Zorn's lemma 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.

References

External links

*MathWorld|urlname=AlgebraicallyIndependent|title=Algebraically Independent|author=Chen, Johnny