Algebraic independence
Set without nontrivial polynomial equalities / From Wikipedia, the free encyclopedia
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In abstract algebra, a subset of a field
is algebraically independent over a subfield
if the elements of
do not satisfy any non-trivial polynomial equation with coefficients in
.
In particular, a one element set is algebraically independent over
if and only if
is transcendental over
. In general, all the elements of an algebraically independent set
over
are by necessity transcendental over
, and over all of the field extensions over
generated by the remaining elements of
.