Frobenius theorem (real division algebras)

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

• Let {{math|D}} be the division algebra in question.
• Let {{math|n}} be the dimension of {{math|D}}.
• We identify the real multiples of {{math|1}} with {{math|R}}.
• When we write {{math|a ≤ 0}} for an element {{mvar|a}} of {{mvar|D}}, we imply that {{mvar|a}} is contained in {{math|R}}.
• We can consider {{mvar|D}} as a finite-dimensional {{math|R}}-vector space. Any element {{mvar|d}} of {{mvar|D}} defines an endomorphism of {{mvar|D}} by left-multiplication, we identify {{mvar|d}} with that endomorphism. Therefore, we can speak about the trace of {{mvar|d}}, and its characteristic- and minimal polynomials.
• For any {{mvar|z}} in {{math|C}} define the following real quadratic polynomial:

::$Q(z;\; x)\; =\; x^2\; -\; 2\backslash operatorname\{Re\}(z)x\; +\; |z|^2\; =\; (x-z)(x-\backslash overline\{z\})\; \backslash in\; \backslash mathbf\{R\}[x].$

:Note that if {{math|z ∈ C ∖ R}} then {{math|Q(z; x)}} is irreducible over {{math|R}}.

The key to the argument is the following

:Claim. The set {{mvar|V}} of all elements {{mvar|a}} of {{mvar|D}} such that {{math|a² ≤ 0}} is a vector subspace of {{mvar|D}} of dimension {{math|n − 1}}. Moreover {{math|D {{=}} R ⊕ V}} as {{math|R}}-vector spaces, which implies that {{mvar|V}} generates {{mvar|D}} as an algebra.

Proof of Claim: Pick {{mvar|a}} in {{mvar|D}} with characteristic polynomial {{math|p(x)}}. By the fundamental theorem of algebra, we can write

:$p(x)\; =\; (x-t\_1)\backslash cdots(x-t\_r)\; (x-z\_1)(x\; -\; \backslash overline\{z\_1\})\; \backslash cdots\; (x-z\_s)(x\; -\; \backslash overline\{z\_s\}),\; \backslash qquad\; t\_i\; \backslash in\; \backslash mathbf\{R\},\; \backslash quad\; z\_j\; \backslash in\; \backslash mathbf\{C\}\; \backslash setminus\; \backslash mathbf\{R\}.$

We can rewrite {{math|p(x)}} in terms of the polynomials {{math|Q(z; x)}}:

:$p(x)\; =\; (x-t\_1)\backslash cdots(x-t\_r)\; Q(z\_1;\; x)\; \backslash cdots\; Q(z\_s;\; x).$

Since {{math|z_j ∈ C ∖ R}}, the polynomials {{math|Q(z_j; x)}} are all irreducible over {{math|R}}. By the Cayley–Hamilton theorem, {{math|p(a) {{=}} 0}} and because {{mvar|D}} is a division algebra, it follows that either {{math|a − t_i {{=}} 0}} for some {{mvar|i}} or that {{math|Q(z_j; a) {{=}} 0}} for some {{mvar|j}}. The first case implies that {{mvar|a}} is real. In the second case, it follows that {{math|Q(z_j; x)}} is the minimal polynomial of {{mvar|a}}. Because {{math|p(x)}} has the same complex roots as the minimal polynomial and because it is real it follows that

:$p(x)\; =\; Q(z\_j;\; x)^k\; =\; \backslash left(x^2\; -\; 2\backslash operatorname\{Re\}(z\_j)\; x\; +\; |z\_j|^2\; \backslash right)^k$

for some {{math|k}}. Since {{math|p(x)}} is the characteristic polynomial of {{mvar|a}} the coefficient of {{math|x^{2k − 1}}} in {{math|p(x)}} is {{math|tr(a)}} up to a sign. Therefore, we read from the above equation we have: {{math|tr(a) {{=}} 0}} if and only if {{math|Re(z_j) {{=}} 0}}, in other words {{math|tr(a) {{=}} 0}} if and only if {{math|a² {{=}} −{{!}}z_j{{!}}² < 0}}.

So {{mvar|V}} is the subset of all {{mvar|a}} with {{math|tr(a) {{=}} 0}}. In particular, it is a vector subspace. The rank–nullity theorem then implies that {{mvar|V}} has dimension {{math|n − 1}} since it is the kernel of $\backslash operatorname\{tr\}\; :\; D\; \backslash to\; \backslash mathbf\{R\}$. Since {{math|R}} and {{mvar|V}} are disjoint (i.e. they satisfy $\backslash mathbf\; R\; \backslash cap\; V\; =\; \backslash \{0\backslash \}$), and their dimensions sum to {{mvar|n}}, we have that {{math|D {{=}} R ⊕ V}}.

For {{math|a, b}} in {{mvar|V}} define {{math|B(a, b) {{=}} (−ab − ba)/2}}. Because of the identity {{math|(a + b)² − a² − b² {{=}} ab + ba}}, it follows that {{math|B(a, b)}} is real. Furthermore, since {{math|a² ≤ 0}}, we have: {{math|B(a, a) > 0}} for {{math|a ≠ 0}}. Thus {{mvar|B}} is a positive-definite symmetric bilinear form, in other words, an inner product on {{mvar|V}}.

Let {{mvar|W}} be a subspace of {{mvar|V}} that generates {{mvar|D}} as an algebra and which is minimal with respect to this property. Let {{math|e₁, ..., e_k}} be an orthonormal basis of {{mvar|W}} with respect to {{math|B}}. Then orthonormality implies that:

:$e\_i^2\; =-1,\; \backslash quad\; e\_i\; e\_j\; =\; -\; e\_j\; e\_i.$

The form of {{mvar|D}} then depends on {{mvar|k}}:

If {{math|k {{=}} 0}}, then {{mvar|D}} is isomorphic to {{math|R}}.

If {{math|k {{=}} 1}}, then {{mvar|D}} is generated by {{math|1}} and {{math|e₁}} subject to the relation {{math|e{{su|b=1|p=2}} {{=}} −1}}. Hence it is isomorphic to {{math|C}}.

If {{math|k {{=}} 2}}, it has been shown above that {{mvar|D}} is generated by {{math|1, e₁, e₂}} subject to the relations

:$e\_1^2\; =\; e\_2^2\; =-1,\; \backslash quad\; e\_1\; e\_2\; =\; -\; e\_2\; e\_1,\; \backslash quad\; (e\_1\; e\_2)(e\_1\; e\_2)\; =-1.$

These are precisely the relations for {{math|H}}.

If {{math|k > 2}}, then {{mvar|D}} cannot be a division algebra. Assume that {{math|k > 2}}. Define {{math|u {{=}} e₁e₂e_k}} and consider {{math|u²{{=}}(e₁e₂e_k)*(e₁e₂e_k)}}. By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that {{math|u² {{=}} 1}}. If {{mvar|D}} were a division algebra, {{math|0 {{=}} u² − 1 {{=}} (u − 1)(u + 1)}} implies {{math|u {{=}} ±1}}, which in turn means: {{math|e_k {{=}} ∓e₁e₂}} and so {{math|e₁, ..., e_k−1}} generate {{mvar|D}}. This contradicts the minimality of {{mvar|W}}.

• The fact that {{mvar|D}} is generated by {{math|e₁, ..., e_k}} subject to the above relations means that {{mvar|D}} is the Clifford algebra of {{math|Rⁿ}}. The last step shows that the only real Clifford algebras which are division algebras are {{math|Cℓ⁰, Cℓ¹}} and {{math|Cℓ²}}.
• As a consequence, the only commutative division algebras are {{math|R}} and {{math|C}}. Also note that {{math|H}} is not a {{math|C}}-algebra. If it were, then the center of {{math|H}} has to contain {{math|C}}, but the center of {{math|H}} is {{math|R}}.
• This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are {{math|R, C, H}}, and the (non-associative) algebra {{math|O}}.
• Pontryagin variant. If {{mvar|D}} is a connected, locally compact division ring, then {{math|D {{=}} R, C}}, or {{math|H}}.

• Hurwitz's theorem, classifying normed real division algebras
• Gelfand–Mazur theorem, classifying complex complete division algebras
• Ostrowski's theorem

• Ray E. Artz (2009) [http://www.math.cmu.edu/~wn0g/noll/qu1.pdf Scalar Algebras and Quaternions], Theorem 7.1 "Frobenius Classification", page 26.
• Ferdinand Georg Frobenius (1878) "[http://commons.wikimedia.org/wiki/File:%C3%9Cber_lineare_Substitutionen_und_bilineare_Formen.djvu 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.

Category:Algebras

Category:Quaternions

Category:Theorems about algebras

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