Multiplicative group of a field ($K^{\times }=K\setminus \{0\}$)

For a field $K$, the nonzero elements form an abelian group under multiplication, called its 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.