Does the group of reals embed into some ultrapower of the rationals?

Question

It is well-known that $\mathbb Q$ and $\mathbb R$ are elementarily equivalent as ordered sets, but how about their group structure? Are they elementarily equivalent as groups? Or more specifically, does $\mathbb R$ embed into some ultrapower of $\mathbb Q$?

Answer 1

Yes, they are elementarily equivalent - this can be shown via Ehrenfeucht-Fraisse games (where we switch from the group structures to their corresponding relational versions - replace the group operation and the inverse operation with the relations defining their graphs). And by the Keisler-Shelah ultrapower theorem, this means that they have isomorphic ultrapowers (so since $\mathbb{R}$ embeds into all its ultrapowers, $\mathbb{R}$ embeds into some ultrapower of $\mathbb{Q}$).

Answer 2

Use a Hamel basis for $\mathbb{R}$ and send each element $r$ of the basis to the hyperrational $\frac{\lfloor{r}{H}\rfloor}{H}$, where $H$ is a fixed infinite hypernatural.

In fact the map $r\mapsto \lfloor{r}{H}\rfloor$ embeds the additive group of the reals into the ultrapower of the integers.