An extension field of a field 
that is not algebraic over 
, i.e., an extension field
that has at least one element that is transcendental over 
.
For example, the field of rational functions 
in the variable 
is a transcendental extension of 
since 
is transcendental over 
. The field 
of real numbers is a transcendental extension of the field
of rational numbers, since 
is transcendental over 
.
