Irreducible
Problems about showing polynomials are irreducible; these are rare in olympiads, but give you a lot of good intuition about how $\mathbb{Z}[x]$ polynomials behave. Techniques that appear include working in $\mathbb{F}_p[x]$, looking at the magnitude of complex roots a la Rouche, and other ad-hoc tricks. More olympiad algebra than the integer polynomials unit.
Irreducible is a rather technical unit of OTIS. The ideas in this unit are not limited to just math Olympiads; this unit can be a reasonable way to get a taste of finite fields and other associated “higher math” topics. The “spark notes” are just half a page long, but don't let appearances deceive you: it takes time to internalize and digest the definitions. This unit is extremely rewarding for developing one's intuition. However, the journey might as well be grueling because typical “Olympiad trick collection” strategy (which is a greedy tactic) won't take you too far.
Philosophy
Let us start by thinking how to think (I wish I could truncate this sentence here) about polynomials with integer coefficients. It is a polynomial ring, denoted by $\mathbb{Z}[X]$. Many-a-times, the polynomial ring corresponding to a ring inherits certain properties from its parent. A good example of this fact is the Hilbert Basis Theorem. So it makes sense to try to find the “analogs” of properties of $\mathbb{Z}$ in $\mathbb{Z}[X]$. We can think of a polynomial:
- In terms of its roots.
- Or, its coefficients.
- As a function of a complex variable
(last one because complex analysis $\gg$ real analysis; anyways, exceptions always exist: you cannot do “inequalities” over complex numbers. You must use $\mathbb{R}$ in some way, which we usually do by employing the absolute value).
$\mathbb{Z}$ is an integral domain (it is a ring and the product of two non-zero elements is non-zero), which is good. But if it were a field, it would have been even better. It is not a field, but we can always make use of the homomorphism $\mathbb{Z}\to\mathbb{F}_p$! From a more algebraic point of view (which means that we are more interested in the roots), finite fields are really cool. If you want a certain element, you can just add it in (like for rational numbers: number fields). But this time, things are much nicer.
For example, if both $2,3$ are non-quadratic residues modulo $p$, then $\mathbb{F}_p[\sqrt{2}]=\mathbb{F}_p[\sqrt{3}]$, which is blatantly false for $\mathbb{Q}$. The key players are the algebraic closure $\overline{\mathbb{F}_p}$ and the Frobenius endomorphism $x\mapsto x^p$, as you shall see. A classic example of the use of the $\mathbb{F}_p$ reduction is the Eisenstein criterion.
Notable Problems
Some problems are classical; some have a distinct algebraic number theoretic flavor. Some are rather hard and worth knowing. Others are simply cool.
- Frobilinear polynomial's irreducible Nice use of the Frobenius map and use of $\overline{\mathbb{F}_p}$.
- ELMO 2012 Problem 3 This is worth knowing in itself; a verification of intuition.
- pseudo-Eisenstein for differences This is a good example of the reduction we talked about in philosophy (because it's not a direct corollary of that; you need to think more).
- Romania TST 2010 A good portrayal of using inequalities and bounding modulus of roots.
- Romania TST 2003 Wonderful example of one algebraic concept$-$ symmetry.