The transcendence degree of 
, sometimes called the transcendental degree, is one because
it is generated by one extra element. In contrast, 
(which is the same field) also has transcendence
degree one because 
is algebraic over 
.
In general, the transcendence degree of an extension
field 
over a field 
is the smallest number elements of 
which are not algebraic over 
, but needed to generate 
. If the smallest set of transcendental
elements needed to generate 
is infinite, then the transcendence degree is the cardinal
number of that set.
For instance, the transcendence degree of 
over 
is one. The transcendence degree of 
over 
is an infinite cardinal number.
There are many open questions about the traditional constants in mathematics, such
as the transcendence degree of 
.
