Good extensions of real numbers

Because complex arithmetic has brought a revolution in mathematics and physics, mathematicians got interested in the steps that extend the field of real numbers R to that of complex numbers C. (C is called a field extension of R, denoted C/R). The idea is to see how far can R be extended and what interesting objects can be obtained. Frobenius proved in 1878 that there are in fact only 3 that are worth interest. His discovery stopped many mathematicians in their new-space-discovering spree.

C shares with R some really neat properties : Those properties are in fact consequences of (C, +) and (C*, ·) being Abelian groups, ie. C is a field.

  • Associativity : x + (y + z) = (x + y) + z and x · (y · z) = (x · y) · z.
  • Commutativity : x + y = y + x and x · y = y · x.
  • Inverse : the additive inverse of x exists and is -x such that x + (-x) = 0 and the multiplicative inverse of x (x ≠ 0) exists and is x-1 such that x · x-1 = 1
  • Distributivity : (x + y) · k = x · k + y · k
  • Cancellation : x · y = 0 implies x = 0 or y = 0

It is impossible to find sets of elements containing R other than R and C that satisfy all of the above axioms so some of them must be omitted. If commutativity can be sacrificed, associativity shouldnt for two good reasons : It makes computations horrible and the physicist could argue that non-associativity does not make sense. He understands that permuting two elements (for example two electronic circuits) is likely to change the result (x · y ≠ y · x), but cannot conceive a way to distinguish x · (y · z) from (x · y) · z.

The idea is to extend C and preserve associativity, distributivity and inverse.

The theorem

Frobenius's theorem : Every division algebra over R of finite dimension is isomorphic to one of the following : the field of real numbers R, the field of complex numbers C or the skew field of quaternions H.

Isomorphic means that there is a one-to-one correspondence between two elements of the respective sets. Say the division algebra A is isomorphic to R, then every element of A can be associated with a real number that behaves the same way. For example let f be such a function :

f : A → R, x → a
∀ x, y ∈ A, f(x+y) = f(x) + f(y) and f(x ·, y) = f(x) · f(y)

This means that all computations in the division algebra can be replaced by simple computations in R, C or H with the following scheme :

∀ x, y ∈ A,
x + y = f-1( f(x) + f(y) ), f(x) + f(y) is an addition in R, C or H
x · y = f-1( f(x) · f(y) ), f(x) · f(y) is a multiplication in R, C or H

Two algebras that are isomorphic are of same dimension. Thus the maximum dimension of an associative division algebra over R is 4.

In other words, only R, C or H are worth something.

Closing words

The proof is fairly simple. Here I outline the basic steps :

  • Prove that every element of a cancellation algebra of finite dimension is either multiple of 1 or satisfies a relation of the form x2 = 2dx + e (e < -d2). Start by showing that it satisfies ax3 + bx2 + cx = 0
  • Prove that every associative cancellation algebra of finite dimension has a basis of the form (1, i1, i2, ... in) where ik2 = -1 and ipiq + iqip = 0, p ≠ q
  • Prove that the basis has no more than 3 i elements in it, and that if it has 2, it has a third one.

As I said above, this discovery checked many mathematicians in their efforts to discover new interesting R-extensions of higher dimension and explains why finding some was so hard. It also shows that it is ok to work with complex numbers, even though you might think that 2 is a rather small dimension for a division algebra.

Sources : MathWorld :
Thanks to krimson for helping me improve this node.