 Multiplicative group of a finite field

The multiplicative group of a finite field ${\mathbb{F}}_q$ is the group ${\mathbb{F}}_q^{\times }$ of its $q-1$ nonzero elements under multiplication. Zero is excluded because it has no multiplicative inverse. In the field with four elements, these elements are $1,\alpha ,\alpha +1$, obtained by repeated multiplication by $\alpha$: they form a cyclic group of group order three. Its addition behaves differently: every nonzero element added to itself gives zero. Keeping the two operations distinct prevents confusing the size of the finite field with the group order of its multiplicative group of a finite field.