Field Homomorphism Preserves Unity

Theorem
Let $\phi: \struct {F_1, +_1, \times_1} \to \struct {F_2, +_2, \times_2}$ be a field homomorphism.

Let:
 * $1_{F_1}$ be the unity of $F_1$
 * $1_{F_2}$ be the unity of $F_2$.

Then:
 * $\map \phi {1_{F_1} } = 1_{F_2}$

Proof
By definition, if $\struct {F_1, +_1, \times_1}$ and $\struct {F_2, +_2, \times_2}$ are fields then $\struct {F_1^*, \times_1}$ and $\struct {F_2^*, \times_2}$ are groups.

Again by definition:
 * the unity of $\struct {F_1, +_1, \times_1}$ is the identity of $\struct {F_1^*, \times_1}$
 * the unity of $\struct {F_2, +_2, \times_2}$ is the identity of $\struct {F_2^*, \times_2}$.

The result follows from Group Homomorphism Preserves Identity.