Editing Transcendental extension (section)

== Facts ==

If ''M'' / ''L'' and ''L'' / ''K'' are field extensions, then

:trdeg(''M'' / ''K'') = trdeg(''M'' / ''L'') + trdeg(''L'' / ''K'')

This is proven by showing that a transcendence basis of ''M'' / ''K'' can be obtained by taking the [[union (set theory)|union]] of a transcendence basis of ''M'' / ''L'' and one of ''L'' / ''K''.

If the set ''S'' is algebraically independent over ''K,'' then the field ''K''(''S'') is [[isomorphic]] to the field of rational functions over ''K'' in a set of variables of the same cardinality as ''S.'' Each such rational function is a fraction of two polynomials in finitely many of those variables, with coefficients in ''K.''

Two [[algebraically closed field]]s are isomorphic if and only if they have the same characteristic and the same transcendence degree over their prime field.<ref>{{harvnb|Milne|loc=Proposition 9.16.}}</ref>