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: R (the real numbers) C (the complex numbers) H (the quaternions) These algebras have real dimension 1, 2 , and 4 , respectively. Of these three algebras, R and C are commutative, but H is not.

Proof

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

Introducing some notation

D be the division algebra in question. n be the dimension of D . 1 with R . a ≤ 0 for an element a of D, we imply that a is contained in R . R -vector space. Any element d of D defines an endomorphism of D by left-multiplication, we identify d with that endomorphism. Therefore, we can speak about the trace of d, and its characteristic- and minimal polynomials. C define the following real quadratic polynomial: z ∈ C ∖ R then Q(z; x) is irreducible over R .

The claim

The key to the argument is the following a2 ≤ 0 is a vector subspace of D of dimension n − 1 . Moreover as R -vector spaces, which implies that V generates D as an algebra. 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, 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. The first case implies that a is real. In the second case, it follows that Q(zj; 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 for some k . Since p(x) is the characteristic polynomial of a the coefficient of x 2k − 1 in p(x) is tr(a) up to a sign. Therefore, we read from the above equation we have: if and only if , in other words if and only if zj . So V is the subset of all a with . In particular, it is a vector subspace. The rank–nullity theorem then implies that V has dimension n − 1 since it is the kernel of. Since R and V are disjoint (i.e. they satisfy ), and their dimensions sum to n, we have that .

The finish

For a, b in V define B(a, b) = (−ab − ba)/2 . Because of the identity (a + b)2 − a2 − b2 = ab + ba , it follows that B(a, b) is real. Furthermore, since a2 ≤ 0 , we have: B(a, a) > 0 for a ≠ 0 . 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. Let e1, ..., ek be an orthonormal basis of W with respect to B . Then orthonormality implies that: The form of D then depends on k: If , then D is isomorphic to R . If , then D is generated by 1 and e1 subject to the relation . Hence it is isomorphic to C . If , it has been shown above that D is generated by 1, e1, e2 subject to the relations These are precisely the relations for H . If k > 2 , then D cannot be a division algebra. Assume that k > 2 . Define and consider . By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that . If D were a division algebra, implies , which in turn means: and so e1, ..., ek−1 generate D. This contradicts the minimality of W.

Remarks and related results

e1, ..., ek subject to the above relations means that D is the Clifford algebra of Rn . The last step shows that the only real Clifford algebras which are division algebras are Cℓ0, Cℓ1 and Cℓ2 . R and C . Also note that H is not a C -algebra. If it were, then the center of H has to contain C , but the center of H is R . R, C, H , and the (non-associative) algebra O . , or H .