Английская Википедия:Frobenius theorem (real division algebras)

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Шаблон:Short description Шаблон:For 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:

These algebras have real dimension Шаблон:Math, and Шаблон:Math, respectively. Of these three algebras, Шаблон:Math and Шаблон:Math are commutative, but Шаблон:Math is not.

Proof

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

Introducing some notation

<math>Q(z; x) = x^2 - 2\operatorname{Re}(z)x + |z|^2 = (x-z)(x-\overline{z}) \in \mathbf{R}[x].</math>
Note that if Шаблон:Math then Шаблон:Math is irreducible over Шаблон:Math.

The claim

The key to the argument is the following

Claim. The set Шаблон:Mvar of all elements Шаблон:Mvar of Шаблон:Mvar such that Шаблон:Math is a vector subspace of Шаблон:Mvar of dimension Шаблон:Math. Moreover Шаблон:Math as Шаблон:Math-vector spaces, which implies that Шаблон:Mvar generates Шаблон:Mvar as an algebra.

Proof of Claim: Pick Шаблон:Mvar in Шаблон:Mvar with characteristic polynomial Шаблон:Math. By the fundamental theorem of algebra, we can write

<math>p(x) = (x-t_1)\cdots(x-t_r) (x-z_1)(x - \overline{z_1}) \cdots (x-z_s)(x - \overline{z_s}), \qquad t_i \in \mathbf{R}, \quad z_j \in \mathbf{C} \setminus \mathbf{R}.</math>

We can rewrite Шаблон:Math in terms of the polynomials Шаблон:Math:

<math>p(x) = (x-t_1)\cdots(x-t_r) Q(z_1; x) \cdots Q(z_s; x).</math>

Since Шаблон:Math, the polynomials Шаблон:Math are all irreducible over Шаблон:Math. By the Cayley–Hamilton theorem, Шаблон:Math and because Шаблон:Mvar is a division algebra, it follows that either Шаблон:Math for some Шаблон:Mvar or that Шаблон:Math for some Шаблон:Mvar. The first case implies that Шаблон:Mvar is real. In the second case, it follows that Шаблон:Math is the minimal polynomial of Шаблон:Mvar. Because Шаблон:Math has the same complex roots as the minimal polynomial and because it is real it follows that

<math>p(x) = Q(z_j; x)^k = \left(x^2 - 2\operatorname{Re}(z_j) x + |z_j|^2 \right)^k</math>

Since Шаблон:Math is the characteristic polynomial of Шаблон:Mvar the coefficient of Шаблон:Math in Шаблон:Math is Шаблон:Math up to a sign. Therefore, we read from the above equation we have: Шаблон:Math if and only if Шаблон:Math, in other words Шаблон:Math if and only if Шаблон:Math.

So Шаблон:Mvar is the subset of all Шаблон:Mvar with Шаблон:Math. In particular, it is a vector subspace. The rank–nullity theorem then implies that Шаблон:Mvar has dimension Шаблон:Math since it is the kernel of <math>\operatorname{tr} : D \to \mathbf{R}</math>. Since Шаблон:Math and Шаблон:Mvar are disjoint (i.e. they satisfy <math>\mathbf R \cap V = \{0\}</math>), and their dimensions sum to Шаблон:Mvar, we have that Шаблон:Math.

The finish

For Шаблон:Math in Шаблон:Mvar define Шаблон:Math. Because of the identity Шаблон:Math, it follows that Шаблон:Math is real. Furthermore, since Шаблон:Math, we have: Шаблон:Math for Шаблон:Math. Thus Шаблон:Mvar is a positive-definite symmetric bilinear form, in other words, an inner product on Шаблон:Mvar.

Let Шаблон:Mvar be a subspace of Шаблон:Mvar that generates Шаблон:Mvar as an algebra and which is minimal with respect to this property. Let Шаблон:Math be an orthonormal basis of Шаблон:Mvar with respect to Шаблон:Math. Then orthonormality implies that:

<math>e_i^2 =-1, \quad e_i e_j = - e_j e_i.</math>

If Шаблон:Math, then Шаблон:Mvar is isomorphic to Шаблон:Math.

If Шаблон:Math, then Шаблон:Mvar is generated by Шаблон:Math and Шаблон:Math subject to the relation Шаблон:Math. Hence it is isomorphic to Шаблон:Math.

If Шаблон:Math, it has been shown above that Шаблон:Mvar is generated by Шаблон:Math subject to the relations

<math>e_1^2 = e_2^2 =-1, \quad e_1 e_2 = - e_2 e_1, \quad (e_1 e_2)(e_1 e_2) =-1.</math>

These are precisely the relations for Шаблон:Math.

If Шаблон:Math, then Шаблон:Mvar cannot be a division algebra. Assume that Шаблон:Math. Let Шаблон:Math. It is easy to see that Шаблон:Math (this only works if Шаблон:Math). If Шаблон:Mvar were a division algebra, Шаблон:Math implies Шаблон:Math, which in turn means: Шаблон:Math and so Шаблон:Math generate Шаблон:Mvar. This contradicts the minimality of Шаблон:Mvar.

Remarks and related results

References

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