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.
The key to the argument is the following Proof of Claim: 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(zj; 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 − ti = 0 for some i or that Q(zj; a) = 0 for some j.
Thus B is a positive-definite symmetric bilinear form, in other words, an inner product on V. Let W be a subspace of V that generates D as an algebra and which is minimal with respect to this property.
Then orthonormality implies that: The form of D then depends on k: If k = 0, then D is isomorphic to R. If k = 1, then D is generated by 1 and e1 subject to the relation e21 = −1.
If D were a division algebra, 0 = u2 − 1 = (u − 1)(u + 1) implies u = ±1, which in turn means: ek = ∓e1e2 and so e1, ..., ek−1 generate D. This contradicts the minimality of W.