I'm trying to understand the proof of the Frobenius's Theorem: "If $A$ is an algebraic division algebra over $\mathbb{R}$ we get $A$ is isomorphic to $\mathbb{R},\mathbb{C},\mathbb{H}$"
The case $\dim A=1$ is trivial. In case $\dim A\geq 2$ the proof takes $x \in A \setminus \mathbb{R}$. Now my question is, what is $A \setminus \mathbb{R}$? It has dimension $1$?
Thank you all!