Frobenius theorem (real division algebras)

In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following:

These algebras have dimensions $1, 2$ , and $4$ , respectively. Of these three algebras, $R$ and $C$ are commutative, but $H$ is not.

The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.

The key to the argument is the following

Proof of Claim: Let $m$ be the dimension of $D$ as an $R$ -vector space, and pick $a$ in $D$ with characteristic polynomial $p (x)$ . By the fundamental theorem of algebra, we can write

We can rewrite $p (x)$ in terms of the polynomials $Q (z; x)$ :

Since $zj ∈ C\R$ , the polynomials $Q (z j; x)$ are all irreducible over $R$ . By the Cayley–Hamilton theorem, $p (a) = 0$ and because $D$ is a division algebra, it follows that either $a - t i = 0$ for some $i$ or that $Q (z j; a) = 0$ for some $j$ . The first case implies that $a$ is real. In the second case, it follows that $Q (z j; x)$ is the minimal polynomial of $a$ . Because $p (x)$ has the same complex roots as the minimal polynomial and because it is real it follows that

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