Frobenius Theorem (real Division Algebras)
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In
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, more specifically in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
division algebras over the
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s. According to the
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, every such algebra is
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to one of the following: * (the real numbers) * (the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s) * (the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s) These algebras have real dimension , and , respectively. Of these three algebras, and are
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, but is not.


Proof

The main ingredients for the following proof are the Cayley–Hamilton theorem and the
fundamental theorem of algebra The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...
.


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 In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
. 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 In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
: ::Q(z; x) = x^2 - 2\operatorname(z)x + , z, ^2 = (x-z)(x-\overline) \in \mathbf :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 :p(x) = (x-t_1)\cdots(x-t_r) (x-z_1)(x - \overline) \cdots (x-z_s)(x - \overline), \qquad t_i \in \mathbf, \quad z_j \in \mathbf \setminus \mathbf. We can rewrite in terms of the polynomials : :p(x) = (x-t_1)\cdots(x-t_r) Q(z_1; x) \cdots Q(z_s; x). 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 A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
as the minimal polynomial and because it is real it follows that :p(x) = Q(z_j; x)^k = \left(x^2 - 2\operatorname(z_j) x + , z_j, ^2 \right)^k 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 logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
, 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 \operatorname : D \to \mathbf. Since and are disjoint (i.e. they satisfy \mathbf R \cap V = \), 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 mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...
, in other words, an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
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 In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
of with respect to . Then orthonormality implies that: :e_i^2 =-1, \quad e_i e_j = - e_j e_i. 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 :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. 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 In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
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 algebra In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-defini ...
s 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 {{ISBN, 0-7923-2459-5 . * Leonard Dickson (1914) ''Linear Algebras'',
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
. 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 ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
75:366–8. * Lev Semenovich Pontryagin, Topological Groups, page 159, 1966. Algebras Quaternions Theorems about algebras Articles containing proofs