complex conjugation

Properties

The map ${\displaystyle \sigma (z)={\overline {z}}}$ from ${\displaystyle \mathbb {C} }$ to ${\displaystyle \mathbb {C} }$ is a homeomorphism (where the topology on ${\displaystyle \mathbb {C} }$ is taken to be the standard topology) and antilinear, if one considers ${\displaystyle \mathbb {C} }$ as a complex vector space over itself. Even though it appears to be a well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a fieldautomorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension⁠${\displaystyle \mathbb {C} /\mathbb {R} }$⁠. This Galois group has only two elements: ${\displaystyle \sigma }$ and the identity on ⁠${\displaystyle \mathbb {C} }$⁠. Thus the only two field automorphisms of ${\displaystyle \mathbb {C} }$ that leave the real numbers fixed are the identity map and complex conjugation.