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: 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. The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra. Let D be the division algebra in question. Let n be the dimension of D. We identify the real multiples of 1 with R. When we write a ≤ 0 for an element a of D, we imply 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 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 dimension n - 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 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. The first case implies that a is real. In the second case, it follows that Q(zj; x) is the minimal polynomial of a.