Frobenius theorem (real division algebras)

Proof

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

Introducing some notation

Let D be the division algebra in question.

We identify the real multiples of 1 with R.

When we write a ≤ 0 for an element a of D, we tacitly assume that a is contained in R.

We can consider D as a finite-dimensional 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.

For any z in C define the following real quadratic polynomial:

Note that if z ∈ C ∖ R then Q(z; x) is irreducible over R.

The claim

The key to the argument is the following

Claim. The set V of all elements a of D such that a² ≤ 0 is a vector subspace of D of codimension 1. Moreover D = R ⊕ V as R-vector spaces, which implies that V generates D as an algebra.

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

p ( x ) = ( x − t 1 ) ⋯ ( x − t r ) ( x − z 1 ) ( x − z 1 ¯ ) ⋯ ( x − z s ) ( x − z s ¯ ) , t i ∈ R , z j ∈ C ∖ R .

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

p ( x ) = ( x − t 1 ) ⋯ ( x − t r ) Q ( z 1 ; x ) ⋯ Q ( z s ; x ) .

Since z_j ∈ CR, 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

p ( x ) = Q ( z j ; x ) k = ( x 2 − 2 Re ⁡ ( z j ) x + | z j | 2 ) 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: tr(a) = 0 if and only if Re(z_j) = 0, in other words tr(a) = 0 if and only if a² = −|z_j|² < 0.

So V is the subset of all a with tr(a) = 0. In particular, it is a vector subspace. Moreover, V has codimension 1 since it is the kernel of a non-zero linear form, and note that D is the direct sum of R and V as vector spaces.

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, ..., e_n be an orthonormal basis of W. With respect to the negative definite bilinear form −B these elements satisfy the following relations:

e i 2 = − 1 , e i e j = − e j e i .

If n = 0, then D is isomorphic to R.

If n = 1, then D is generated by 1 and e₁ subject to the relation e2
1 = −1. Hence it is isomorphic to C.

If n = 2, it has been shown above that D is generated by 1, e₁, e₂ subject to the relations

e 1 2 = e 2 2 = − 1 , e 1 e 2 = − e 2 e 1 , ( e 1 e 2 ) ( e 1 e 2 ) = − 1.

These are precisely the relations for H.

If n > 2, then D cannot be a division algebra. Assume that n > 2. Let u = e₁e₂e_n. It is easy to see that u² = 1 (this only works if n > 2). If D were a division algebra, 0 = u² − 1 = (u − 1)(u + 1) implies u = ±1, which in turn means: e_n = ∓e₁e₂ and so e₁, ..., e_n−1 generate D. This contradicts the minimality of W.

Remarks and related results

The fact that D is generated by e₁, ..., e_n 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.

As a consequence, the only commutative division algebras are 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. Therefore, the only division algebra over C is C itself.

This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are R, C, H, and the (non-associative) algebra O.

Pontryagin variant. If D is a connected, locally compact division ring, then D = R, C, or H.