# Algebraically independent

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*, that is, if for every finite sequence α

_{1},...,α

_{n} of elements of

*S*, no two the same, and every non-zero polynomial

*P*(

*x*_{1},...,

*x*_{n}) with coefficients in

*K*, we have

*P*(α_{1},...,α_{n})≠0.

In particular, a one element set {

*α*} is algebraically independent over

*K* if and only if α is

transcendental over

*K*.