= Multiplicative group of a finite field
{title2=$\mathbb F_q^\times$}
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>.
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