Dan Shved

My mathematics blog

Frobenius theorem on real division algebras

This post is a proof of the well known Frobenius theorem. It’s nothing groundbreaking, I’m just writing it down to organize my own thoughts.

Definition. A real division algebra is a set A together with operations +: A \times A \to A, \cdot : \mathbb {R}\times A \to A and \ast : A \times A \to A such that:

  1. (A, +, \cdot ) is a vector space over \mathbb {R}.
  2. (A, +, \ast ) is a division ring.
  3. r \cdot (a_1 \ast a_2) = (r \cdot a_1) \ast a_2 = a_1 \ast (r \cdot a_1) for every r \in \mathbb {R} and a_1, a_2 \in A.

Examples of real division algebras: \mathbb {R} itself, the algebra of complex numbers \mathbb {C}, the algebra of quaternions \mathbb {H}.

Theorem (Frobenius). Every finite-dimensional real division algebra is isomorphic to one of \mathbb {R}, \mathbb {C} or \mathbb {H}.

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An exercise in transfinite magic

In my hometown there is a university called SUSU. And in this university regular math contests (link in Russian) are held, organized by A. Yu. Evnin. The following problem appeared in one of these contests:

Problem 1. A function f: \mathbb {R} \to \mathbb {R} has the following property: every straight line on the plane \mathbb {R}^2 intersects the graph of f and the parabola y = x^2 at the same number of points. Prove that f(x) = x^2 for every x \in \mathbb {R}.

The solution is quite straightforward, I am not going to write it down. It is probably enough to say that it heavily exploits the convexity of y = x^2.

This post is not about problem 1. It is about the next thing that comes to mind:

Problem 2. Two functions f, g: \mathbb {R} \to \mathbb {R} have graphs F and G such that for every straight line l \subset \mathbb {R}^2: |l \cap F| = |l \cap G| < \infty. Does it follow that f and g are the same function?

If you think about this one for a minute, I’m sure you’ll find it much more challenging than problem 1. This whole post is about solving problem 2. If you want to give it a try yourself, this is the last point in the text where you can still do this without knowing the answer.

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A problem with polynomial functions

Here is a problem that popped up in my head about a year ago.

Problem 1. A function f: \mathbb {Z} \times \mathbb {Z} \to \mathbb {Z} is polynomial of each argument, i.e. for every x_0 \in \mathbb {Z} the function y \to f(x_0, y) can be represented by a polynomial with integer coefficients, and the same goes for x \to f(x, y_0) for every y_0. Is it true then that f must be polynomial of both arguments simultaneously, i.e. representable by an element of \mathbb {Z}[x,y]?

If you’ve ever been a freshman in an analysis course, you are bound to be aware of a common trap with the notion of differentiability. The trap is like this: you are asked to prove that a function g: \mathbb {R} \times \mathbb {R} \to \mathbb {R} is differentiable everywhere. You have several months of single-variable analysis behind your back. So your first instinct may be to prove that \partial g / \partial x and \partial g / \partial y exist at every point and be done with it. But this is not enough, as demonstrated by

\displaystyle g(x,y) = \left\{ \begin{array}{ll}0 &  \quad \mathrm{at}\, \mathrm{point}\, (0,0) \\ \frac{xy}{\sqrt {x^2+y^2}} &  \quad \mathrm{everywhere}\, \mathrm{else.}\end{array}\right.

Problem 1 asks if there is the same trap with polynomiality (not a word?) as there is with differentiability. When I came up with problem 1 I was busy with more “serious” math, so the question didn’t get any real thought. I always suspected that it would’t take long to solve, but somehow I was too lazy to do the job. That is, I was until this morning, when I finally put in the required several minutes and got this thing sorted out. If you’re curious, you might want to do the same before moving on to the next paragraph.

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Getting started

After all, I’m starting a blog. The blog will be about mathematics. If this takes off and doesn’t fade away after three posts, I’ll probably make an “About” page that will explain what kind of a blog this is going to be.

The general idea is to have a place where I could keep my papers, as well as proofs and ideas that could be interesting but cannot be published in a journal for some reason.

Constructive criticism is very welcome, both about mathematics and about the language in general. If you see something in a post that looks like a grammar mistake (for instance) — please do e-mail me: danshved [at] gmail.com. This kind of feedback is very welcome (since I am an English learner).

GPG key: E8FEE1E6.