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(
x1,...,
xn) with coefficients in
K, we have
- P(α1,...,αn)≠0.
In particular, a one element set {
α} is algebraically independent over
K if and only if α is
transcendental over
K.