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:

$\forall z \in \C: \map \phi z = \overline z$

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

By Sum of Complex Conjugates:

$\map \phi {z_1 + z_2} = \map \phi {z_1} + \map \phi {z_2}$

By Product of Complex Conjugates:

$\map \phi {z_1 z_2} = \map \phi {z_1} \map \phi {z_2}$

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.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms: Example $5$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): automorphism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): automorphism