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(x₁, ..., x_n) with coefficients in K, we have Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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. In mathematics, a transcendental function is a function which is not expressible as a composition of a finite number of elementary operations, or inverses of functions so constructible, where the elementary operations consist of addition, multiplication, taking additive or multiplicative inverses, and integer root extraction. ...

For example, the subset {√π, 2π + 1} of R is not algebraically independent over the integers, since the non-zero polynomial Lower-case pi The mathematical constant Ï€ is a real number which is defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

P(x₁, x₂) = 2x₁² − x₂ + 1

yields zero when √π is substituted for x₁ and 2π + 1 is substituted for x₂.

It is not known whether the set {π,e} is algebraically independent over Q. Nesterenko proved in 1996 that {π, e^π, Γ(1/4)} is algebraically independent over Q. The mathematical constant e is the base of the natural logarithm. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non natural numbers (where it is defined). ...

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 these maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every partially ordered set in which every chain (i. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the size of the extension. ...