Algebraic independence

In abstract algebra, a subset ${\displaystyle S}$ of a field ${\displaystyle L}$ is algebraically independent over a subfield ${\displaystyle K}$ if the elements of ${\displaystyle S}$ do not satisfy any non-trivial polynomial equation with coefficients in ${\displaystyle K}$.

In particular, a one element set ${\displaystyle \{\alpha \}}$ is algebraically independent over ${\displaystyle K}$ if and only if ${\displaystyle \alpha }$ is transcendental over ${\displaystyle K}$. In general, any element of an algebraically independent set ${\displaystyle S}$ over ${\displaystyle K}$ is by necessity transcendental over ${\displaystyle K}$, and over all of the field extensions of ${\displaystyle K}$ generated by the remaining elements of ${\displaystyle S}$.