The following definition is analogous to the characteristic of a ring:
Let $(F, + ,\cdot)$ be a field, let $1\in F$ be its neutral element of multiplication $"\cdot",$ and let $0\in F$ be its neutral element of addition $"+".$
A characteristic of a field $\operatorname{char}( F )$ is the minimal1 natural number \(n\), for which \[\underbrace{1 + \ldots + 1}_{n\text{ times}}=n\cdot 1=0.\]
Such minimal element exists due to the well-ordering principle of the natural numbers. If in a given field \(F\) there is no such such $n$ with $n > 0$, then $n=0$ will do the trick. ↩
