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 this 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 Fields , , , and the p-adic Number have characteristic 0. For a Prime, the Galois Field GF() has characteristic .
If is a Subfield of , then and have the same characteristic.
See also Field, Subfield