The following definition is analogous to the characteristic of a ring:

Definition: Characteristic of a Field

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 \[\underbrace{1 + \ldots + 1}_{n\text{ times}}=n\cdot 1=0.\]

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  1. 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.