Characteristic three ($\mathrm{char}K=3$)

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.