Algebraic independence

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

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

Example

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

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

${\displaystyle P(x,y)=2x^{2}-y+1}$

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

Algebraic independence of known constants

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

• the numbers ${\displaystyle \pi }$, ${\displaystyle e^{\pi }}$, and ${\displaystyle \Gamma (1/4)}$, where ${\displaystyle \Gamma }$ is the gamma function, are algebraically independent over ${\displaystyle \mathbb {Q} }$. ^[3]
• the numbers ${\displaystyle e^{\pi {\sqrt {3}}}}$ and ${\displaystyle \Gamma (1/3)}$ are algebraically independent over ${\displaystyle \mathbb {Q} }$.
• for all positive integers ${\displaystyle n}$, the number ${\displaystyle e^{\pi {\sqrt {n}}}}$ is algebraically independent over ${\displaystyle \mathbb {Q} }$. ^[4]

Lindemann–Weierstrass theorem

The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over ${\displaystyle \mathbb {Q} }$. It states that whenever ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$ are algebraic numbers that are linearly independent over ${\displaystyle \mathbb {Q} }$, then ${\displaystyle e^{\alpha _{1}},\ldots ,e^{\alpha _{n}}}$ are also algebraically independent over ${\displaystyle \mathbb {Q} }$.

Algebraic matroids

Main article: Algebraic matroid

Given a field extension ${\displaystyle L/K}$ that is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of ${\displaystyle L}$ over ${\displaystyle K}$. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.

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

1. Patrick Morandi (1996). Field and Galois Theory. Springer. p. 174. ISBN   978-0-387-94753-2 . Retrieved April 11, 2008.
2. Green, Ben (2008), "III.41 Irrational and Transcendental Numbers", in Gowers, Timothy (ed.), The Princeton Companion to Mathematics , Princeton University Press, p. 222
3. Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). p. 61. ISBN   978-3-540-20364-3. ISSN   0938-0396. Zbl   1079.11002.
4. Nesterenko, Yuri V (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I . 322 (10): 909–914.
5. Ingleton, A. W.; Main, R. A. (1975), "Non-algebraic matroids exist", Bulletin of the London Mathematical Society , 7 (2): 144–146, doi:10.1112/blms/7.2.144, MR   0369110 .
6. Joshi, K. D. (1997), Applied Discrete Structures, New Age International, p. 909, ISBN   9788122408263 .

External links

• Chen, Johnny. "Algebraically Independent". MathWorld .