For a field 
with 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 number 
, 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, the finite
field GF(
)
has characteristic 
.
If 
is a subfield of 
, then 
and 
have the same characteristic.
