Formally, a subset S of E is algebraically independent over F if any polynomial with coefficients in F, say f(X1,...,Xn), such that f(s1,...,sn)=0 where the si are distinct elements of S, must be zero as a polynomial.
If there is a non-zero polynomial f such that f(s1,...,sn)=0, then the si are said to be algebraically dependent.
Any subset of an algebraically independent set is algebraically independent.
An algebraically independent subset of E of maximal cardinality is a transcendence basis for E/F, and this cardinality is the transcendence degree or transcendence dimension of E over F.
Algebraic independence has the exchange property: if G is a set such that E is algebraic over F(G), and I is a subset of G which is algebraically independent, then there is a subset B of G with which is a transcendence basis. The algebraically independent subsets thus form an independence structure.