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,
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.
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
∀ 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.
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 : http://mathworld.wolfram.com/
Thanks to krimson for helping me improve this node.