= Multiplicative group of a field
{title2=$K^\times=K\setminus\{0\}$}
= Field multiplicative group
{synonym}
For a <field> $K$, the nonzero elements form an <abelian group> under multiplication, called its <multiplicative group of a field>[multiplicative group]. Products of nonzero elements are nonzero because a <field> has no nonzero <zero divisors>; multiplication is associative and commutative, the identity is $1$, and every nonzero element has an inverse. Thus every finite <subgroup> of $K^\times$ is a <finite abelian group>. The <finite subgroup of a field multiplicative group is cyclic> theorem uses the <root bound for a polynomial> to prove a stronger restriction on such subgroups.
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