New Zealand Maths Olympiad Committee online

Round about sixth form one learns that every polynomial can be factorized, as a product of linear factors.  Why?  Well, here’s a polynomial, see.  It’s probably a cubic with integer coefficients — after all, most nontrivial polynomials that one encounters are.  You play with it until you discover a root, likely by looking at integer factors of the highest and lowest coefficients.  Then you polynomial-divide through by the linear factor which that root gives you, and get a quadratic, whose roots there’s a formula for finding.  Tada!

Of course, there’s a problem with this algorithm:  it depends on figuring out how to break down your polynomial into only linear and quadratic factors.

Perhaps you know (at least in theory) the formulae for roots of general cubics and quartics.  Then you can carry out the above algorithm even if initially you can only see how to break down your polynomial into linear, quadratic, cubic and quartic factors.  That certainly covers most of the polynomials you’ll ever encounter at school.

But then problems arise.  The Fundamental Theorem of Algebra is the aforementioned factorization theorem.  More precisely, it states that every polynomial with integer, rational, real, and, importantly, even complex coefficients has a unique decomposition into linear complex-coefficiented factors.  Its proof is one of the coolest things you’ll learn in university complex analysis.  On the other hand, Galois theory, one of the coolest things you’ll learn in university abstract algebra, shows that there’s no general formula for the roots of greater-than-fourth degree polynomials.  The result is the kind of paradox that logicians love to hate:  polynomials with roots that you know exist, but which you have no algorithm for constructing.

The two links in the paragraph above are to sample articles of the Princeton Companion to Mathematics, a recently-published “encyclopedia of modern pure mathematics” edited by mathematician, Fields medallist, British 1981 IMO perfect scorer, and blogger Timothy Gowers.  More sample articles available here (log in with username Guest and password PCM, and click the link to Resources).  Two others which I especially like are: a quick tour of fundamental definitions and structures (Section Four is a great introduction to the notions of limit and continuity, which, as an Olympiad hopeful, you may well have struggled through lately); and a discussion of how professional mathematicians go about problem-solving.

Olympiad mathematics is full of beautiful ideas, which show up all the better for being restricted to within a tightly-contained syllabus.  Modern pure mathematics, as the Companion does an exhilarating job of showing, takes the beautiful ideas and lets them loose.