Characteristic of a field

Summary

The characteristic of a field is the least positive number of copies of $1$ that add up to $0$. It is $0$ when no such number exists. It is the characteristic of the field as a ring, and for a field it is always $0$ or a prime number. It decides the prime subfield, and many results of field theory depend on it. For example, the map $a \mapsto a^p$ is a ring homomorphism in characteristic $p$.