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In [[abstract algebra]], the '''transcendence degree''' of a [[field extension]] ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest [[cardinality]] of an [[algebraically independent]] subset of ''L'' over ''K''.
A subset ''S'' of ''L'' is a '''transcendence basis''' of ''L'' / ''K'' if it is algebraically independent over ''K'' and if furthermore ''L'' is an [[algebraic extension]] of the field ''K''(''S'') obtained by adjoining the elements of ''S'' to ''K''. One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension.
If no field ''K'' is specified, the transcendence degree of a field ''L'' is its degree relative to the prime field of the same [[characteristic]], i.e., '''Q''' if ''L'' is of characteristic 0 and '''F'''<sub>''p''</sub> if ''L'' is of characteristic ''p''.
The field extension ''L'' / ''K'' is '''purely transcendental''' if there is a subset ''S'' of ''L'' that's algebraically independent over ''K'' and such that ''L'' = ''K''(''S'').
== Examples ==
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