# 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*(*X*_{1},...,*X*_{n}), such that *f*(*s*_{1},...,*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*(*s*_{1},...,*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 which is a transcendence basis. The algebraically independent subsets thus form an independence structure.

## 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.