A field Isomorphism $f:\mathbb{C} \rightarrow \mathbb{C}$

Question

Any suggestions for a field Isomorphism $f:\mathbb{C} \rightarrow \mathbb{C}$ other than the identity map?

Answer 1

How about the map $\phi:\Bbb C\to\Bbb C$ given by $\phi(z) = \overline{z}$? It satisfies $\phi(0) = 0$, $\phi(1) = 1$, $\phi(z_1+z_2) = \phi(z_1)+\phi(z_2)$ and $\phi(z_1z_2) = \phi(z_1)\phi(z_2)$.

Answer 2

Hint: Define $f: \mathbb C \to \mathbb C$ by $f(z) = \overline {z} $.

Answer 3

Complex field automorphisms other than the identity and the complex conjugation do exist, and are called "wild". They are described in the expository paper by Paul B. Yale, Automorphisms of the complex numbers, Math. Mag. 39 (1966), 135-141.

Answer 4

The only continuous automorphisms of $\mathbf C$ are the identity map and conjugation.

However, using the axiom of choice, it can be shown there is a infinity of non-continuous field automorphisms of $\mathbf C$. You can look at this paper.