I’ll preface this post with the fact that I’m assuming a working knowledge of abstract algebra at the undergraduate level, realistically more about definitions than theorems. I’m taking a class this semester on Galois theory and I’ve been occasionally prepping throughout the summer to get back up to speed in various abstract algebra topics. The textbook, Galois Theory (2nd. edition) by Joseph Rotman, is pretty dense compared to what I’m used to; it (reasonably) assumes a familiarity with groups and jumps right into reviewing commutative rings, starting with the definition,

A few weeks ago, I was mindlessly browsing the Mathematics Stack Exchange when I came across a really interesting question (that I unfortuntately can no longer locate); how do you sort a real-valued continuous function? The initial naive approach is straightforward: take some $f: \mathbb{R} \to \mathbb{R}$ that’s continuous and, just like in typical countable sets, create a sequence $\mathcal{F}$ where $\mathcal{F}_0 = \inf(f)$ and