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?

or of Field isomorphism of $\mathbb{C}$ onto itself. Seriously, try searching. — , Mar 01 '16 at 23:15

score 7 · Answer 1 · edited Mar 01 '16 at 23:13

7

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.

edited Mar 01 '16 at 23:13

answered Mar 01 '16 at 22:07

C. Dubussy

9,373

score 4 · Accepted Answer · answered Mar 01 '16 at 22:07

4

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 $\phi(z_1z_2) = \phi(z_1)\phi(z_2)$.

answered Mar 01 '16 at 22:07

Cameron L. Williams

30,485

score 3 · Answer 3 · answered Mar 01 '16 at 22:07

3

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

answered Mar 01 '16 at 22:07

Aaron Maroja

17,869

score 2 · Answer 4 · answered Mar 01 '16 at 22:23

2

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.

answered Mar 01 '16 at 22:23

Bernard

179,256

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

4 Answers4