Monsky’s Theorem – one square and odd number of triangles

In the previous post, I discussed valuation theory and promised to give a cool application. Here we shall ask ourselves a question that is easy to understand and seems to have no relation at all to number theory: Given an odd number $latex n$, is it possible to dissect a square into $latex n$ triangles …

Absolute Values and Valuation Theory

In this post, we will generalise the familiar absolute value function and discuss the dual concept of valuation. I promise that a surprising application of the theory we discussed here will be shared in the next post! Basic Facts About Absolute Value Definition 1: Let $latex F$ be a field. An absolute value on $latex …

Constructible numbers; Compass and Straightedge constructions

Simply put, a real number $latex r$ is constructible if, starting form a line segment of unit length, a line segment of length $latex |r|$ can be constructed with a compass and straightedge in a fintie number of steps. The study of constructible numbers is (elegantly) linked to 4 famous problems in Euclidean geometry: (1) …

Transcendental Numbers

This post is a short write-up on transcendental numbers. Generally speaking, given an extension of fields $latex F \subseteq E$ and an element $latex \alpha \in E$, we say that $latex \alpha$ is algebraic over $latex F$ if $latex \alpha$ satisfies some nonzero polynomial $latex f\in F[X]$. Otherwise, we say that $latex \alpha$ is transcendental …