Multiplicative Group of Field is Abelian Group/Proof 2

Theorem

Let $\struct {F, +, \times}$ be a field.

Let $F^* := F \setminus \set 0$ be the set $F$ less its zero.

The algebraic structure $\struct {F^*, \times}$ is an abelian group.

Proof

Recall that a field is a non-trivial commutative division ring.

The result follows from Non-Zero Elements of Division Ring form Group.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55.3$ Special types of ring and ring elements