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