The transcendence degree of an extension field 
over a field 
is the cardinal
number of a transcendence basis, a maximal algebraically
independent subset 
over 
. Equivalently, 
is algebraically
independent and 
is an algebraic extension
of the field 
generated by 
and 
. The cardinal number is
independent of the choice of 
.
For example, 
has transcendence degree one over 
, as does 
, which is the same field.
The definition allows 
to contain further algebraic
elements over 
and does not require 
alone to generate 
as a field.
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 
.
