Automorphism

Remarks

Example: the map $z \mapsto \bar{z}$ is an automorphism of the field $(\C, +, \cdot, 0, 1)$, because it is a bijection of $\C$ to itself with $\overline{z + w} = \bar{z} + \bar{w}$, $\overline{zw} = \bar{z}\,\bar{w}$, $\bar{0} = 0$ and $\bar{1} = 1$.

Non-example: the map $x \mapsto 2x$ on $(\Z, +, 0)$ is an injective homomorphism but not an automorphism, because it is not surjective: $1$ is not in its image.

The set $\Aut(\mathbf{A})$ is a group under composition, with the identity map as identity element. The only automorphism of the field $\R$ is the identity.