Complex Conjugation is Automorphism

Theorem
Consider the Field of Complex Numbers $$\C$$.

The operation of complex conjugation:
 * $$\forall z \in \C: z \mapsto \overline z$$

is a field automorphism.

Proof
Let $$z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$$.

Let us define the mapping $$\phi: \C \to \C$$ defined as:
 * $$\forall z \in \C: \phi \left({z}\right) = \overline z$$

We check that $$\phi$$ has the morphism property:

$$ $$ $$ $$ $$

$$ $$ $$ $$ $$

So the morphism property holds for both complex addition and complex multiplication.

Hence we can say that complex conjugation is a field homomorphism.

We note that $$\overline z_1 = \overline z_2 \implies z_1 = z_2$$ and so complex conjugation is injective.

Also, complex conjugation is trivially surjective, and hence bijective.

The result then follows by definition of field automorphism.