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

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.

| | | | | created: 2019-08-10 07:12:33 | modified: 2019-08-10 07:21:08 | by: bookofproofs