# 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 x
^{2} = 2dx + e (e < -d^{2}).
Start by showing that it satisfies ax^{3} + bx^{2} + cx = 0
- Prove that every associative cancellation algebra of finite dimension has a basis of the form
(1, i
_{1}, i_{2}, ... i_{n}) where i_{k}^{2} = -1 and
i_{p}i_{q} + i_{q}i_{p} = 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.