This is the crispest proof I’ve found that the complex numbers ℂ are the only finite field extension of the real numbers, ℝ.
In other words, if you want to continue the process which takes you from the number line ℝ to the number plane ℂ, you can’t. Once you’ve generalised from counting numbers to including zero, to negative numbers, to fractions, to irrational numbers, and then to complex numbers, you’re done. Number systems don’t keep on generalising. Complex numbers are it.
You can define an addition and multiplication on points in a plane which makes them obey all the rules of arithmetic which hold for the number line – that is, more or less, what it means to call that structure a field – but there’s no way of doing that with points in 3D (or 4D, or 5D… ) space.
You can define addition and multiplication on points in 4D space which obeys most of the rules of arithmetic which hold for the number line, and that creates a structure called quaternions, much used these days, so I’m told, in designing graphics for computer games. But in quaternions a.b=−b.a.
The proof is from Abstract Algebra, by Celine Carstensen, Benjamin Fine, and Gerhard Rosenberger (Walter de Gruyter, 2011), p.262 (and pp.186-7 for the Sylow theorems).
It proves the result together with the “Fundamental Theorem of Algebra”, namely, that in ℂ every polynomial has solutions, which also has a neat topological proof.
A field extension is a bigger field which differs from the smaller field in including solutions to polynomial equations insoluble in the smaller field.
If K is a field extension of ℝ, then there is another mathematical structure, K:ℝ, which is the vector space formed by K over ℝ. For example, ℂ forms a two-dimensional vector space over ℝ – each point in ℂ requires two numbers to describe it.
The dimension of K:ℝ can’t be odd, since polynomials of odd degree always have a solution in ℝ, and therefore the minimal polynomial of K, the polynomial of smallest degree which generates it, must be of even degree.
Suppose K is a field extension of ℂ. Then we look at another structure, the Galois group of K over ℝ, which is all the ways that the elements of K can be mapped onto each other while keeping the arithmetic structure of K and while mapping every element of ℝ into itself. (For example, mapping every complex number to its conjugate is a member of the Galois group of ℂ over ℝ).
Then the order (“size”) of the Galois group must be 2m.q for some odd number q. A standard theorem of group theory, one of Sylow’s theorems, then tells us that this group has a subgroup of order 2m. But that would be the Galois group of a smaller extension E with dimension of E:ℝ=q, which is impossible (from the above) unless q=1.
Another of Sylow’s theorems tells that the Galois group of K over ℂ which is of order 2m−1, must have a subgroup of order 2m−2, which corresponds to an intermediate field D which is a degree 2 extension of ℂ. But, no field can be a degree 2 extension of ℂ, because by the quadratic formula every quadratic in ℂ is soluble.
Therefore the finite field extension K is ℂ itself (or an identical copy of it). ∎
Mathematicians use lots of other number systems, and call them “number systems” – transfinite numbers, surreal numbers, p-adic numbers, and much more – and there are also infinite field extensions of ℝ – but in the direct sense ℂ is the end of the process of expanding our sense of number which we first embark on when we realise the counting numbers continue indefinitely.