The Hamiltonian ring is not a complex algebra

The Hamiltonian algebra ( contains isomorphicPlanetmathPlanetmathPlanetmath copies of the real and complex numbers. However, the reals are a central subalgebra of which makes into a real algebra. This makes identifying in canonical: 1 determines a unique embeddingMathworldPlanetmathPlanetmath :rr1. Yet is not a complex algebra. The goal presently is to outline some of the incongruities of =1,i and =1,ı^,ȷ^,k^ which may be obscured by the notational overlap of the letter i.

Proposition 1.

There are no proper finite dimensional division rings over algebraically closed fields.


Let D be a finite dimensional division ring over an algebraically closed field K. This means that K is a central subalgebra of D. Let aD and consider K(a). Since K is central in D, K(a) is commutativePlanetmathPlanetmathPlanetmath, and so K(a) is a field extension of K. But as D is a finite dimensional K space, so is K(a). As any finite dimensional extensionPlanetmathPlanetmathPlanetmath of K is algebraic, K(a) is an algebraic extensionMathworldPlanetmath. Yet K is algebraically closedMathworldPlanetmath so K(a)=K. Thus aK so in fact D=K. ∎

  • In particular, this propositionPlanetmathPlanetmath proves is not a complex algebra.

  • Alternatively, from the Wedderburn-Artin theorem we know the only semisimplePlanetmathPlanetmathPlanetmathPlanetmath complex algebra of dimensionMathworldPlanetmath 2 is . This has proper idealsMathworldPlanetmath and so it cannot be the division ring .

  • It is also evident that the usual, notationally driven, embedding of into is non-central. That is, embeds as a+bia+bi^, into =1,ı^,ȷ^,k^. This is not central:

  • Further evidence of the incompatiblity of and comes from considering polynomialsPlanetmathPlanetmath. If x2+1 is considered as a polynomial over [x] then it has exactly two roots i,-i as expected. However, if it is considered as a polynomial over [x] we arrive at 6 obvious roots: {ı^,-ı^,ȷ^,-ȷ^,k^,-k^}. But indeed, given any q, q0, then qı^q-1 is also a root. Thus there are an infiniteMathworldPlanetmathPlanetmath number of roots to x2+1. Therefore declaring ı^=-1 can be greatly misleading. Such a conflict does not arise for polynomials with real roots since is a central subalgebra.

Title The Hamiltonian ring is not a complex algebra
Canonical name TheHamiltonianRingIsNotAComplexAlgebra
Date of creation 2013-03-22 16:01:57
Last modified on 2013-03-22 16:01:57
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 10
Author Algeboy (12884)
Entry type Result
Classification msc 16W99