Negative of Element in Field is Unique

Theorem

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

Let $a \in F$.

Then the negative $-a$ of $a$ is unique.

Proof 1

By definition, a field is a ring whose ring product less zero is an abelian group.

The result follows from Ring Negative is Unique.

$\blacksquare$

Proof 2

Let $b, c \in F$ such that both $a + b = 0$ and $a + c = 0$.

Thus:

\(\ds b + \paren {a + c}\)

\(=\)

\(\ds b + 0\)

as $c$ is a negative of $a$

\(\ds \)

\(=\)

\(\ds b\)

Definition of Field Zero

But also:

\(\ds \paren {b + a} + c\)

\(=\)

\(\ds \paren {a + b} + c\)

Field Axiom $\text A1$: Associativity of Addition

\(\ds \)

\(=\)

\(\ds 0 + c\)

as $b$ is a negative of $a$

\(\ds \)

\(=\)

\(\ds c\)

Definition of Field Zero

So $b = c$ and the result follows.

$\blacksquare$