Endomorphisms of $\mathbb{H}$ that do not fix $\mathbb{R}$

Question

Are there any nontrivial field endomorphisms of $\mathbb{H}$ that are not the identity map when restricted to $\mathbb{R}$? I realize that the only nontrivial endomorphism on $\mathbb{R}$ is the identity map, so if the desired function $f$ exists, it must be that $f(\mathbb{R}) \ne \mathbb{R}$. Are there any such mappings?

Answer 1

There are no such automorphisms of $\Bbb{H}$. The argument from an earlier answer of mine implies that any ring homomorphism $f:\Bbb{H}\to\Bbb{H}$

• is surjective (this is non-trivial),
• is injective (a division ring has no non-trivial ideals, so this is trivial), and
• hence maps the center $\Bbb{R}$ to itself.

You also already knew that the identity is the only endomorphism of $\Bbb{R}$, so the question is settled.