# Multiplicative Group of Field is Abelian Group/Proof 2

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## 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