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In abstract algebra, 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 $K$ .

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