characteristic of a field

Start from 1 and keep adding it to itself: 1, 1+1, 1+1+1, and so on. In the rationals you never come back to 0, but in arithmetic modulo a prime p you do — after p steps the sum wraps around to zero. The characteristic of a field is precisely how many such steps it takes to reach 0, or the verdict 'never'. It is a single number, 0 or a prime, that controls the deepest behavior of the field.

Precisely, the characteristic of a field K is the least positive integer n with n times 1 (the n-fold sum 1 + 1 + ... + 1) equal to 0, and is defined to be 0 if no such n exists. The characteristic is always either 0 or a prime p, because if it were a composite n = ab then ab = 0 with neither factor zero would contradict that a field has no zero divisors.

Characteristic splits field theory into two worlds. Characteristic 0 fields contain a copy of the rationals and are always perfect; characteristic p fields contain a copy of F_p, possess the Frobenius endomorphism x to x^p, and can harbor inseparability. The smallest subfield generated by 1 — the prime field — is determined entirely by the characteristic: Q in characteristic 0, F_p in characteristic p.