## Any Automorphism of the Field of Real Numbers Must be the Identity Map

## Problem 507

Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve laterProve that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve laterLet $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers.

Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers.

Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups.