Frobenius's Theorem

Theorem
An algebraic associative real division algebra $A$ is isomorphic to $\R, \C$ or $\Bbb H$.

Proof
Next we recall that an algebra A is said to be quadratic if it is unital and the elements 1; x; x 2 are linearly dependent for every x 2 A. Thus, for every x 2 A there exist t(x); n(x) 2 R such that x 2 t(x)x + n(x) = 0. Obviously, t(x) and n(x) are uniquely determined if x =2 R. Setting t(�) = 2� and n(�) = � 2 for � 2 R, we can then consider t and n as maps from A into R (the reason for this deﬁnition is that in this way t becomes a linear functional, but we shall not need this). We call t(x) and n(x) the trace and the norm of x, respectively. For some elementary properties of quadratic algebras, a characterization of quadratic alternative algebras, and further references we refer to [9]. From x 2 (x+x � )x+x � x = 0 we see that all algebras An are quadratic. Further, every real division algebra A that is algebraic and power-associative (this means that every subalgebra generated by one element is associative) is automatically quadratic. Indeed, if x 2 A then there exists a nonzero polynomial f(X) 2 R[X] such that f(x) = 0. Writing f(X) as the product of linear and quadratic polynomials in R[X] it follows that p(x) = 0 for some p(X) 2 R[X] of degree 1 or 2. In particular, algebraic alternative (and hence associative) real division algebras are quadratic. Finally, if A is a real unital algebra, i.e., an algebra over R with unity 1, then we shall follow a standard convention and identify R with R1; thus we shall write � for �1, where � 2 R.

Lemma 3.1 Let A be a quadratic real algebra. Then U = fu 2 AnR j u 2 2 Rg[f0g is a linear subspace of A, uv + vu 2 R for all u; v 2 U, and A = R � U. Proof. Obviously, U is closed under scalar multiplication. We have to show that u; v 2 U implies u + v 2 U. If u; v; 1 are linearly dependent, then one easily notices that already u and v are dependent, and the result follows. Thus, let u; v; 1 be independent. We have (u + v) 2 + (u v) 2 = 2u 2 + 2v 2 2 R. On the other hand, as A is quadratic there exist �; � 2 R such that (u + v) 2 �(u + v) 2 R and (uv) 2�(uv) 2 R, and hence �(u+v)+�(uv) 2 R. However, the independence of 1; u; v implies � + � = � � = 0, so that � = � = 0. This proves that u � v 2 U. Thus U is indeed a subspace of A. Accordingly, uv +vu = (u+v) 2 u 2 v 2 2 R for all u; v 2 U. Finally, if a 2 A n R, then a 2 �a 2 R for some � 2 R, and therefore u = a � 2 2 U; thus, a = � 2 + u 2 R � U. �

Remark 3.2. If A is additionally a division algebra, then every nonzero u 2 U can be written as u = �v with � 2 R and v 2 = 1. Indeed, since u 2 2 R and since u 2 cannot be � 0 (otherwise (u �)(u + �) = u 2  � 2 would be 0 for some � 2 R) we have u 2 = � 2 with 0 =6 � 2 R. Thus, v = � 1 u is a desired element.

Lemma 3.3 Let A be a quadratic real division algebra, and let U be as in Lemma 3.1. Suppose e1; : : : ; ek 2 U are such that e 2 i = 1 for all i � k and eiej = ej ei for all i; j � k, i =6 j. If U is not equal to the linear span of e1; : : : ; ek, then there exists ek+1 2 U such that e 2 k+1 = 1 and eiek+1 = ek+1ei for all i � k.

Proof. Pick u 2 U that is not contained in the linear span of e1; : : : ; ek, and set �i = 1 2 (uei +eiu) 2 R (by Lemma 3.1). Note that v = u+�1e1 +: : :+�kek satisﬁes eiv = vei for all i � k. Let ek+1 be a scalar multiple of v such that e 2 k+1 = 1 (Remark 3.2). Then ek+1 has all desired properties.

As pointed out at the end of Section 2, A is quadratic. We may assume that n = dim A � 2. By Remark 3.2 we can ﬁx i 2 A such that i 2 = 1. Thus, A =� C if n = 2. Let n > 2. By Lemma 3.3 there is j 2 A such that j 2 = 1 and ij = ji. Set k = ij. Now one immediately checks that k 2 = 1, ki = j = ik, jk = i = kj, and i; j; k are linearly independent. Therefore A contains a subalgebra isomorphic to H. It remains to show that n is not > 4. If it was, then by Lemma 3.3 there would exist e 2 A such that e = 0 6, ei = ie, ej = je, and ek = ke. However, from the ﬁrst two identities we infer eij = iej = ije; since ij = k, this contradicts the third identity.