Zero of Field is Unique

Theorem

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

The zero of $F$ 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 Zero is Unique.

$\blacksquare$

Proof 2

Let $0_1$ and $0_2$ both be elements of $F$ such that:

$\forall a \in F: a + 0_1 = a$

$\forall a \in F: a + 0_2 = a$

Then:

$0_1 + 0_2 = 0_2$

because $0_1$ is a zero element

$0_1 + 0_2 = 0_1$

because $0_2$ is a zero element

Hence:

$0_1 = 0_2$

and the two zero elements are the same.

$\blacksquare$