# 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 dimensions 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

- We identify the real multiples of 1 with
**R**.

- When we write
*a*≤ 0 for an element Template:Mvar of Template:Mvar, we tacitly assume that Template:Mvar is contained in**R**.

- We can consider Template:Mvar as a finite-dimensional
**R**-vector space. Any element Template:Mvar of Template:Mvar defines an endomorphism of Template:Mvar by left-multiplication, we identify Template:Mvar with that endomorphism. Therefore we can speak about the trace of Template:Mvar, and its characteristic and minimal polynomials.

- For any Template:Mvar in
**C**define the following real quadratic polynomial:

### The claim

The key to the argument is the following

**Claim.**The set Template:Mvar of all elements Template:Mvar of Template:Mvar such that*a*^{2}≤ 0 is a vector subspace of Template:Mvar of codimension 1. Moreover*D*=**R**⊕*V*as**R**-vector spaces, which implies that Template:Mvar generated Template:Mvar as an algebra.

**Proof of Claim:** Let Template:Mvar be the dimension of Template:Mvar as an **R**-vector space, and pick Template:Mvar in Template:Mvar 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 *z _{j}* ∈

**C**\

**R**, the polynomials

*Q*(

*z*;

_{j}*x*) are all irreducible over

**R**. By the Cayley–Hamilton theorem,

*p*(

*a*) = 0 and because Template:Mvar is a division algebra, it follows that either

*a*−

*t*= 0 for some Template:Mvar or that

_{i}*Q*(

*z*;

_{j}*a*) = 0 for some Template:Mvar. The first case implies that Template:Mvar is real. In the second case, it follows that

*Q*(

*z*;

_{j}*x*) is the minimal polynomial of Template:Mvar. Because

*p*(

*x*) has the same complex roots as the minimal polynomial and because it is real it follows that

Since *p*(*x*) is the characteristic polynomial of Template:Mvar 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

*a*

^{2}= −|

*z*|

_{j}^{2}< 0.

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

### The finish

For *a*, *b* in Template:Mvar define *B*(*a*, *b*) = (−*ab* − *ba*)/2. Because of the identity (*a* + *b*)^{2} − *a*^{2} − *b*^{2} = *ab* + *ba*, it follows that *B*(*a*, *b*) is real. Furthermore since *a*^{2} ≤ 0, we have: *B*(*a*, *a*) > 0 for *a* ≠ 0. Thus Template:Mvar is a positive definite symmetric bilinear form, in other words, an inner product on Template:Mvar.

Let Template:Mvar be a subspace of Template:Mvar that generates Template:Mvar as an algebra and which is minimal with respect to this property. Let *e*_{1}, ..., *e _{n}* be an orthonormal basis of Template:Mvar. With respect to the negative definite bilinear form −

*B*these elements satisfy the following relations:

If *n* = 0, then Template:Mvar is isomorphic to **R**.

If *n* = 1, then Template:Mvar is generated by 1 and *e*_{1} subject to the relation *e*Template:Su = −1. Hence it is isomorphic to **C**.

If *n* = 2, it has been shown above that Template:Mvar is generated by 1, *e*_{1}, *e*_{2} subject to the relations

These are precisely the relations for **H**.

If *n* > 2, then Template:Mvar cannot be a division algebra. Assume that *n* > 2. Let *u* = *e*_{1}*e*_{2}*e _{n}*. It is easy to see that

*u*

^{2}= 1 (this only works if

*n*> 2). If Template:Mvar were a division algebra, 0 =

*u*

^{2}− 1 = (

*u*− 1)(

*u*+ 1) implies

*u*= ±1, which in turn means:

*e*= ∓

_{n}*e*

_{1}

*e*

_{2}and so

*e*

_{1}, ...,

*e*

_{n−1}generate Template:Mvar. This contradicts the minimality of Template:Mvar.

- The fact that Template:Mvar is generated by
*e*_{1}, ...,*e*subject to the above relations means that Template:Mvar is the Clifford algebra of_{n}**R**^{n}. The last step shows that the only real Clifford algebras which are division algebras are Cℓ^{0}, Cℓ^{1}and Cℓ^{2}.

- 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 Template:Mvar is a connected, locally compact division ring, then*D*=**R**,**C**, or**H**.

## 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 ISBN 0-7923-2459-5 . - 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.