Course Summary

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern algebra, the subject of Math 305, seems to break from this theme, instead focusing on axiomatic systems that encompass some familiar algebraic objects, but without much reference to solving polynomials. In this course, we show how the tools of modern algebra really do connect back to the backbone of algebra by developing Galois theory.

In its purest form, Galois theory is an attempt to create a catalog of intermediate fields for a given field extension. More concretely, this means if we're given fields $E$ and $F$ with $F \subseteq E$, Galois theory seeks to find an object which parameterizes all those fields $L$ with $F \subseteq L \subseteq E$. On the surface, this problem doesn't seem to be related to solutions to polynomials, but we'll see throughout the course that field extensions are intimately connected to solutions to (irreducible) polynomials, and that finding "nice" solutions to those polynomials can be neatly described in terms of how "easily" one can "climb" from $F$ to $E$.

In addition to being an incredibly beautiful theory in its own right, the tools of Galois theory can be used to solve a number of classical "unsolvability" problems, particularly the impossibility of certain classical ruler-and-compass construction problems as well as the nonexistence of a quintic analog of the famous quadratic equation.

Aside from the topic of this course, it's worth noting that the pedagogical approach to this particular class is different than most: students will be responsible for the vast majority of the lecturing in the course.

Course Instructor

The professor for this class is Andy Schultz. His office is on the main floor of Clapp Library, room 255. Office hours will be held

You are highly encouraged to attend office hours, and you never need an appointment to do so. If these office hours don't fit with your schedule, contact the instructor so that he can either adjust when ``official" office hours are held or set up an appointment to help you outside of office hours. Please come to the professor's office or send him an email if you ever want to discuss material from the class or ask about homework problems!

You can contact the instructor at . Though he is always happy to receive emails from you with questions or concerns about the course, he can't guarantee that he'll be able to promptly reply to emails late at night or over the weekend. If you do contact the professor by email, please be sure to follow standard email etiquette. In particular, please make sure you include a greeting and signature and avoid abbreviations. If you're contacting him to ask about a problem, please be sure to specify what the problem asks (as opposed to asking something like ``I can't get problem 2 and need your help").