Zero of Subfield is Zero of Field/Proof 1

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0$.

Let $\struct {K, +, \times}$ be a subfield of $\struct {F, +, \times}$.

The zero of $\struct {K, +, \times}$ is also $0$.

Proof

By definition, $\struct {F, +, \times}$ and $\struct {K, +, \times}$ are both rings.

Thus $\struct {K, +, \times}$ is a subring of $\struct {F, +, \times}$

The result follows from Zero of Subring is Zero of Ring.

$\blacksquare$