# Unity of Subfield is Unity of Field

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## Theorem

Let $\struct {F, +, \times}$ be a field whose unity is $1$.

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

The unity of $\struct {K, +, \times}$ is also $1$.

## Proof

By definition, $\struct {K, +, \times}$ is a subset of $F$ which is a field.

By definition of field, $\struct {K^*, \times}$ and $\struct {F^*, \times}$ are groups such that $K \subseteq F$.

So $\struct {K^*, \times}$ is a subgroup of $\struct {F^*, \times}$.

By Identity of Subgroup, the identity of $\struct {F^*, \times}$, which is $1$, is also the identity of $\struct {K^*, \times}$.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 56.3$ Subrings and subfields