The field characteristic of a field 
is determined by repeated addition
of its multiplicative identity 1. Consider
the numbers 
, 
, 
, etc. Either these numbers are all different,
in which case we say that 
has characteristic 0, or two of them will be equal.
In the latter case, it is straightforward to show that, for
some positive integer 
, we have 
. If 
is chosen to be as small as possible,
then 
will be a prime, and we say that 
has characteristic 
. The characteristic of a field 
is sometimes denoted 
.
The fields 
(rationals), 
(reals), 
(complex numbers), and the
p-adic numbers 
have characteristic 0. For 
a prime and 
a positive integer, the
finite field GF(
) has characteristic 
.
If 
is a subfield of 
, then 
and 
have the same characteristic.
