Preface A Note to Students
Welcome! We are so glad you are here.
This book was written in response to a bit of student feedback from one of the author’s (Janssen) first time teaching his institution’s introduction to abstract algebra. In short, the feedback was: I am a future teacher, and I do not know why I had to take this course. The criticism was fair; much of what we had covered that semester did not look like what one typically thinks of as “algebra”, yet it definitely was. In the intervening years, we set out to rethink the way we introduce students to this beautiful subject, and subsequently developed this free resource to do so.
The focus of the first three chapters is factorization, the ability to write certain objects (e.g., numbers, polynomials) as a product of simpler objects (such as prime numbers). This is something you have likely been doing for several years, but we have generally taken it as an article of faith that this can be done in a unique way. The first three chapters of this work has the goal of uncovering structural conditions sufficient to guarantee something like the unique factorization into primes that we know and love from the integers. The fourth chapter is a coda that allows for a deeper exploration of our main objects of interest (rings), as well as functions that relate these objects to one another.
This book was written with the belief that the best way to learn mathematics is to do mathematics. Thus, there are vanishingly few worked examples or proved theorems. Instead, you will provide the proofs and solutions to exercises. It is possible, if you look hard enough, to find some of the answers elsewhere on the internet. I implore you to resist the urge to do so. You will learn much more by struggling with a problem, even if you do not ultimately solve it, than you will by searching for an answer after a few minutes of toying with a problem. The rewards of your struggle will be deep and long-lasting.
Let’s begin.
