Algebraically independent

From Exampleproblems

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

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 set {√π, 2π + 1} is not algebraically independent, since the non-zero polynomial

P(x₁, x₂) = x₁² − 1/2 x₂ + 1/2

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

It is unknown 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.

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.