Classify all finite-dimensional (associative) division algebras over the field of real numbers \(\mathbb{R}\) (Frobenius’ Theorem).
Solution
Frobenius’ theorem. Every finite-dimensional associative division algebra over \(\mathbb R\) is isomorphic to exactly one of \[ \mathbb R,\qquad \mathbb C,\qquad \mathbb H. \]
One standard proof uses the Brauer group. Let \(D\) be such a division algebra and let \(Z=Z(D)\). The center is a finite field extension of \(\mathbb R\), hence \[ Z\cong\mathbb R\quad\text{or}\quad Z\cong\mathbb C. \] If \(Z=\mathbb C\), then \(D=\mathbb C\): for any \(x\in D\), the finite-dimensional commutative subalgebra \(\mathbb C[x]\) is a field, so the minimal polynomial of \(x\) over the algebraically closed field \(\mathbb C\) must be linear.
It remains to consider central division algebras over \(\mathbb R\). The standard computation \[ \operatorname{Br}(\mathbb R)\cong\mathbb Z/2\mathbb Z \] has two classes, represented by \(\mathbb R\) and the Hamilton quaternion algebra \(\mathbb H\). Every central simple algebra over \(\mathbb R\) is therefore Morita-equivalent to one of these; its division-algebra representative is respectively \(\mathbb R\) or \(\mathbb H\). Hence a noncommutative finite-dimensional real division algebra is \(\mathbb H\).
Thus the complete list is \[ \boxed{\mathbb R,\ \mathbb C,\ \mathbb H}. \]