Field Homomorphism Preserves Product Inverses

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

Then:
 * $\forall x \in F_1^*: \map \phi {x^{-1} } = \map \phi x^{-1}$

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 product inverse of $x$ in $\struct {F_1, +_1, \times_1}$ for $\times_1$ is the product inverse of $x$ in $\struct {F_1^*, \times_1}$
 * the product inverse of $x$ in $\struct {F_2, +_2, \times_2}$ for $\times_2$ is the product inverse of $x$ in $\struct {F_2^*, \times_2}$

The result follows from Group Homomorphism Preserves Inverses.