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:

By Sum of Complex Conjugates:
 * $\phi \left({z_1 + z_2}\right) = \phi \left({z_1}\right) + \phi \left({z_2}\right)$

By Product of Complex Conjugates:
 * $\phi \left({z_1 z_2}\right) = \phi \left({z_1}\right) \phi \left({z_2}\right)$

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.