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.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms