Characteristic of a field ($\mathrm{char}F$)

The characteristic of a field $F$ is the least positive integer $n$ for which $n\cdot 1_F=0$, or zero if no such integer exists.

A field has characteristic three when $1+1+1=0$. The binomial identity then becomes $(a+b{)}^3=a^3+b^3$, and $X^3-1=(X-1{)}^3$. Thus a cyclic shift operator on three coordinates has a repeated-root minimal polynomial in this characteristic and cannot be diagonalizable.

In a field of characteristic two, $1+1=0$. The usual splitting of a tensor product into its symmetric square and exterior square by $(I\pm \tau )/2$ is unavailable.

A field has characteristic zero when no positive integer multiple of its identity vanishes. Equivalently, its prime field is isomorphic to $\mathbb{Q}$. Every irreducible polynomial over such a field is a separable polynomial.