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: (the real numbers), (the complex numbers), or (the quaternions). These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but is not.
Proof The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.
Introducing some notation Let be the division algebra in question. Let be the dimension of . We identify the real multiples of with . When we write for an element of , we imply that is contained in . We can consider as a finite-dimensional -vector space. Any element of defines an endomorphism of by left-multiplication, we identify with that endomorphism. Therefore, we can speak about the trace of , and its characteristic- and minimal polynomials. For any in define the following real quadratic polynomial:
Note that if then is irreducible over .
The claim The key to the argument is the following
Claim. The set of all elements of such that is a vector subspace of of dimension . Moreover as -vector spaces, which implies that generates as an algebra.
Proof of Claim: Pick in with characteristic polynomial . By the fundamental theorem of algebra, we can write
We can rewrite in terms of the polynomials :
Since , the polynomials are all irreducible over . By the Cayley–Hamilton theorem, and because is a division algebra, it follows that either for some or that for some . The first case implies that is real. In the second case, it follows that is the minimal polynomial of . Because has the same complex roots as the minimal polynomial and because it is real it follows that
for some . Since is the characteristic polynomial of the coefficient of in is up to a sign. Therefore, we read from the above equation we have: if and only if , in other words if and only if .
So is the subset of all with . In particular, it is a vector subspace. The rank–nullity theorem then implies that has dimension since it is the kernel of . Since and are disjoint (i.e. they satisfy ), and their dimensions sum to , we have that .
The finish For in define . Because of the identity , it follows that is real. Furthermore, since , we have: for . Thus is a positive-definite symmetric bilinear form, in other words, an inner product on .
Let be a subspace of that generates as an algebra and which is minimal with respect to this property. Let be an orthonormal basis of with respect to . Then orthonormality implies that:
The form of then depends on :
If , then is isomorphic to .
If , then is generated by and subject to the relation . Hence it is isomorphic to .
If , it has been shown above that is generated by subject to the relations
These are precisely the relations for .
If , then cannot be a division algebra. Assume that . Define and consider . By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that . If were a division algebra, implies , which in turn means: and so generate . This contradicts the minimality of .
Remarks and related results The fact that is generated by subject to the above relations means that is the Clifford algebra of . The last step shows that the only real Clifford algebras which are division algebras are and . As a consequence, the only commutative division algebras are and . Also note that is not a -algebra. If it were, then the center of has to contain , but the center of is .
This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are , and the (non-associative) algebra . Pontryagin variant. If is a connected, locally compact division ring, then , or . See also Hurwitz's theorem, classifying normed real division algebras Gelfand–Mazur theorem, classifying complex complete division algebras Ostrowski's theorem
References Ray E. Artz (2009) Scalar Algebras and Quaternions, Theorem 7.1 "Frobenius Classification", page 26. Ferdinand Georg Frobenius (1878) "Über lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1–63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp. 343–405. Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp. 30–2 . Leonard Dickson (1914) Linear Algebras, Cambridge University Press. See §11 "Algebra of real quaternions; its unique place among algebras", pages 10 to 12. R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366–8. Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.