View source for Frobenius theorem (real division algebras)

{{short description|Theorem in abstract algebra}}
{{For|other theorems named after Frobenius|Frobenius theorem (disambiguation){{!}}Frobenius theorem}}
In [[mathematics]], more specifically in [[abstract algebra]], the '''Frobenius theorem''', [[mathematical proof|proved]] by [[Ferdinand Georg Frobenius]] in 1877, characterizes the [[dimension (vector space)|finite-dimensional]] [[associative algebra|associative]] [[division algebra]]s over the [[real number]]s.  According to the [[theorem]], every such algebra is [[isomorphic]] to one of the following:
* {{math|'''R'''}} (the real numbers)
* {{math|'''C'''}} (the [[complex number]]s)
* {{math|'''H'''}} (the [[quaternion]]s)
These algebras have real dimension {{math|1, 2}}, and {{math|4}}, respectively.  Of these three algebras, {{math|'''R'''}} and {{math|'''C'''}} are [[commutative algebra (structure)|commutative]], but {{math|'''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===
* 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 (linear algebra)|trace]] of {{mvar|d}}, and its [[characteristic polynomial|characteristic]]- and [[minimal polynomial (linear algebra)|minimal polynomial]]s.
* For any {{mvar|z}} in {{math|'''C'''}} define the following real [[quadratic polynomial|quadratic]] [[polynomial]]:
::<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 {{math|''z'' ∈ '''C'''&thinsp;∖&thinsp;'''R'''}} then {{math|''Q''(''z''; ''x'')}} is [[irreducible polynomial|irreducible]] over {{math|'''R'''}}.

===The claim===
The key to the argument is the following

:'''Claim.''' The set {{mvar|V}} of all elements {{mvar|a}} of {{mvar|D}} such that {{math|''a''<sup>2</sup> ≤ 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

:<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 {{math|''p''(''x'')}} in terms of the polynomials {{math|''Q''(''z''; ''x'')}}:

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

Since {{math|''z<sub>j</sub>'' ∈ '''C'''&thinsp;∖&thinsp;'''R'''}}, the polynomials {{math|''Q''(''z<sub>j</sub>''; ''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<sub>i</sub>'' {{=}} 0}} for some {{mvar|i}} or that {{math|''Q''(''z<sub>j</sub>''; ''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<sub>j</sub>''; ''x'')}} is the minimal polynomial of {{mvar|a}}. Because {{math|''p''(''x'')}} has the same complex [[root of a polynomial|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>

for some {{math|''k''}}. Since {{math|''p''(''x'')}} is the characteristic polynomial of {{mvar|a}} the coefficient of {{math|''x''<sup>&hairsp;2''k'' −&thinsp;1</sup>}} 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<sub>j</sub>'') {{=}} 0}}, in other words {{math|tr(''a'') {{=}} 0}} if and only if {{math|''a''<sup>2</sup> {{=}} −{{!}}''z<sub>j</sub>''{{!}}<sup>2</sup> < 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 (linear algebra)|kernel]] of <math>\operatorname{tr} : D \to \mathbf{R}</math>. Since {{math|'''R'''}} and {{mvar|V}} are disjoint (i.e. they satisfy <math>\mathbf R \cap V = \{0\}</math>), and their dimensions sum to {{mvar|n}}, we have that {{math|''D'' {{=}} '''R''' ⊕ ''V''}}.

===The finish===
For {{math|''a'', ''b''}} in {{mvar|V}} define {{math|''B''(''a'', ''b'') {{=}} (−''ab'' − ''ba'')/2}}. Because of the identity {{math|(''a'' + ''b'')<sup>2</sup> − ''a''<sup>2</sup> − ''b''<sup>2</sup> {{=}} ''ab'' + ''ba''}}, it follows that {{math|''B''(''a'', ''b'')}} is real. Furthermore, since {{math|''a''<sup>2</sup> ≤ 0}}, we have: {{math|''B''(''a'', ''a'') > 0}} for {{math|''a'' ≠ 0}}. Thus {{mvar|B}} is a [[definite bilinear form|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|''e''<sub>1</sub>, ..., ''e<sub>k</sub>''}} be an [[orthonormal basis]] of {{mvar|W}} with respect to {{math|''B''}}. Then orthonormality implies that:

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

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|''e''<sub>1</sub>}} 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, ''e''<sub>1</sub>, ''e''<sub>2</sub>}} 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 {{math|'''H'''}}.

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

==Remarks and related results==
*The fact that {{mvar|D}} is generated by {{math|''e''<sub>1</sub>, ..., ''e<sub>k</sub>''}} subject to the above relations means that {{mvar|D}} is the [[Clifford algebra]] of {{math|'''R'''<sup>''n''</sup>}}. The last step shows that the only real Clifford algebras which are division algebras are {{math|Cℓ<sup>0</sup>, Cℓ<sup>1</sup>}} and {{math|Cℓ<sup>2</sup>}}.
*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 (ring theory)|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 (normed division algebras)|Hurwitz's theorem]], which states that the only real [[normed division algebra]]s are {{math|'''R''', '''C''', '''H'''}}, and the (non-associative) algebra {{math|[[octonion|'''O''']]}}.
* '''Pontryagin variant.''' If {{mvar|D}} is a [[connected space|connected]], [[locally compact space|locally compact]] division [[topological ring|ring]], then {{math|''D'' {{=}} '''R''', '''C'''}}, or {{math|'''H'''}}.
==See also==
* [[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]], classifying normed real division algebras
* [[Gelfand–Mazur theorem]], classifying complex complete division algebras
* [[Ostrowski's theorem#Another Ostrowski's theorem|Ostrowski's theorem]]

==References==
* 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.&nbsp;343–405.
* Yuri Bahturin (1993) ''Basic Structures of Modern Algebra'', Kluwer Acad. Pub. pp.&nbsp;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]], [[List of publications in mathematics#Topological Groups|Topological Groups]], page 159, 1966.

[[Category:Algebras]]
[[Category:Quaternions]]
[[Category:Theorems about algebras]]
[[Category:Articles containing proofs]]