Field

We say that is a field if:

• is an Abelian Group with identity
• is an Abelian Group
• Distributivity holds:

Theorem

Every finite Subgroup of the multiplicative Group of a Field is cyclic.

Proof

Suppose is a finite multiplicative subgroup of a field.
Then is Abelian.
Hence, contains a subgroup isomorphic to for some .
But then has at least roots.