Notes
This is where I place some of my auxiliary and expository notes on random topics in mathematics that interest me. I attempt to keep these notes as self contained as possible, however, they often assume some familiarity with topics typically covered in undergraduate mathematics courses. I will list some reccomended prerequisites for each note. This section collects auxiliary and expository notes on mathematics.
The tricky details not mentioned in the “Pi hiding in prime regularities” video
In Grant’s video, a floor function disappears from a sum and we end up with an infinite series. This seems innocent enough, but the errors could add up and bite you in the butt. Here I work through why they don’t.
Recommended prerequisites: basic divisibility, infinite series (especially alternating series), and limits.
Irreducible polynomials over finite fields
The main point of this short note is to outline exactly why \(x^{p^n}-x\) is the product of all the irreducible polynomials in \(\mathbb{F}_p[x]\) of degree dividing \(n\). At the time I wrote these notes I was unable to find a complete and understandable (from the perspective of an undergraduate student) proof of this statement online. Once we have proven the statement we can then easily obtain a function that gives us the exact number of irreducible polynomials of any degree in \(\mathbb{F}_p[x]\) for any prime \(p\) using the magical Möbius inversion formula.
Recommended prerequisites: polynomial rings and irreducibility, basic finite fields, field extensions and the tower law. Familiarity with Möbius inversion is helpful for the final counting formula.