Algebraic independence

In algebra, algebraic independence is a property of a set of elements of an extension field E/F, that they satisfy no non-trivial algebraic relation.

Formally, a subset S of E is algebraically independent over F if any polynomial with coefficients in F, say f(X₁,...,X_n), such that f(s₁,...,s_n)=0 where the s_i are distinct elements of S, must be zero as a polynomial.

If there is a non-zero polynomial f such that f(s₁,...,s_n)=0, then the s_i 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 $I\subseteq B\subseteq G$ which is a transcendence basis. The algebraically independent subsets thus form the independent sets of a matroid.

Examples

• The singleton set {s} is algebraically independent if and only if s is transcendental over F.

References

• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 355-357. ISBN 0-201-55540-9.