It is well known that you cannot divide by zero, lest you be doomed to triviality. Conversely, in a field, you can divide by anything except zero. But this rule can be misleading, since it's possible that (even) an ordinary number can be zero when you don't expect it! The characteristic of a field states when (if ever) this happens.
Let be a rig (possibly a ring, possibly a commutative ring, possibly even a field). Then there exists a unique homomorphism to from the initial rig, which is the rig of natural numbers. The kernel of is an ideal of , which (by a well-known property of ) is a principal ideal with a unique generator. This generator is the characteristic of , denoted .
If is a ring, then we may use instead, where is the ring of integers. However, in this case, the kernel will usually have two generators, in which case we pick the positive one to get the same result as above.
Every rig with positive characteristic is in fact a ring, since we have . In other words, any rig other than a ring must have characteristic zero (although many rings also have that characteristic).
If is a positive natural number, then the characteristic of is . This rig is always a commutative ring, and it is a field if and only if is prime, in which case it is the prime field? . More generally, every finite field has positive prime characteristic.
For , , , and the prime field (the field of rational numbers) are no longer all the same, but they still all have characteristic . Every ordered field has characteristic . The real numbers and complex numbers each form fields of characteristic .