Inner product spaces, I – Real closed fields

I’ve always felt sort of uneasy about the way linear algebra is presented: you start off doing all this stuff that makes complete sense, and works over arbitrary fields, and then suddenly you’re doing something with all these complex conjugates and conjugate transposes and real symmetric matrices and so forth and nothing makes sense any more.  So I’m going to try to say something about that here.

Everything in the theory of inner products is based on three properties that look simple enough at first glance, but appear more and more bizarre as you consider them more deeply:

  • The real numbers are an ordered field.
  • The real numbers aren’t algebraically closed, but their algebraic closure (the complex numbers) forms a degree-2 extension.
  • The norm of a nonzero complex number is a positive real number.

(By the way, what do we mean by the norm here?  Well, it’s probably exactly what you think, namely

N(z) = z \overline{z}.

But the norm is something more general: if we have a finite Galois extension L/K, then we can define a function N_{L/K} : L \to K by

N_{L/K}(a) := \prod\limits_{\sigma \in {\rm Gal}(L/K)} \sigma(a)

 Since {\rm Gal}(\mathbb{C}/\mathbb{R}) consists of the identity and complex conjugation, we recover

N_{\mathbb{C}/\mathbb{R}}(z) = z \overline{z}.)

The fact that N_{\mathbb{C}/\mathbb{R}}(z) > 0 for z \neq 0 is specific to $latex \mathbb{C}/\mathbb{R}$; for instance, if d is some squarefree positive integer, then we have

N_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}(1 + \sqrt{d}) = (1 + \sqrt{d})(1 - \sqrt{d}) = 1-d.

Anyway, here’s why this is all pretty weird:

  • For almost any other field, the algebraic closure is an infinite-dimensional extension, so we have no hope of getting a norm map like this.  In fact, if we have a field F whose algebraic closure \overline{F} is a finite-dimensional extension, then F is a real closed field, meaning that it looks very much like the real numbers, and moreover \overline{F} = F[i].  (This is the Artin-Schreier theorem.)
  • In particular, if F is an ordered field then N_{F[i]/F}(x + i y) = (x + i y)(x - i y) = x^2 + y^2 > 0 for x + i y \neq 0.
  • So, there seem to be two completely separate kinds of non-algebraically closed fields: those that behave exactly like this (such as the real algebraic numbers, reals, and the field of real Puiseux series), and those that behave nothing like this but much like one another (such as the rational numbers, number fields in general, positive characteristic fields, etc.).

The fact that we have (1) an ordered field R (2) whose algebraic closure C is a finite-degree extension  such that (3) N_{R/C}(z) > 0 for nonzero z \in C allows us to extend the theory of linear algebra (over both R and C!) in some strange new directions.

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