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.
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 
. 
(rationals),
(reals), 
(complex numbers), and the p-adic numbers 
have characteristic 0. For a prime, the finite field GF(
) has characteristic 
. 
is a subfield of 
, then 
and 
have the same characteristic.