Instructor: Nicholas Katz
Description of Course Goals and CurriculumThe course is an introduction to abstract algebra which is the study of mathematical structures such as groups, rings, and fields with their applications. The theoretical component centers around developing the framework of Galois theory which is used to study fields; the overarching applied component is to answer a classical question in mathematics: is every polynomial solvable in radicals? For example, given a quadratic, there is a quadratic formula to find the roots; does the same apply for all polynomials of higher degree? This question is introduced in the first week of lecture and answered in the last: in between, Katz focuses on developing the tools required to address such an abstract problem. The course has two fast-paced lectures per week whose ideas are reinforced by the problem sets. The problem sets are collaborative and open-note; the midterm and final are both open-note but not collaborative. The problem sets typically cover material from the previous week whereas the midterm and final are lengthy and cumulative.
Learning From Classroom InstructionKatz's lectures are very dense and fast-paced - meaning that being lost for a second in class can quickly spiral out of control. Katz focuses on doing proofs in lecture so it is often difficult to extract the essentials for problem sets and future classes. Because it is easy to lose sight of the focus of the course, Katz's teaching assistants are invaluable for those confused about homework or lecture topics. Most topics in the class are cumulative, so it is very hard to understand lectures or even start the problems without an understanding of the previous lecture material. Typically, the teaching assistants will point out the big ideas behind the theorems and the essential parts of proofs which can be hard to identify when first introduced. The only textbook for the class is the Artin's "Algebra with Galois Theory" which is terse and therefore difficult to parse at times. Katz never "assigns" readings, but skimming Artin before class is helpful to understand the overarching ideas from the lectures. Although not officially recommended by Katz, Dummit and Foote's "Abstract Algebra" is the standard text when other professors teach the course. By nature of being a longer textbook, it has insightful explanations of ideas from class with worked-out examples and great practice problems. I found Dummit and Foote invaluable for understanding material post-lecture.
Learning For and From AssignmentsFor the problem sets, I would recommend skimming over the problems and then reading the relevant portions of Dummit and Foote before attempting to solve them. If there is still confusion about the material, the TA office hours are helpful when one comes prepared with questions. Even "I don't understand what I should be learning here" can prompt helpful responses. Problem sets are returned quickly and it is best to review them as soon as possible. It is helpful to have in mind the thought process behind your solution and compare it to what the graders had in mind. Also regarding problem sets, I would recommend working in a small group. It is helpful to exchange ideas with classmates and if everybody is confused about one topic, it is clear that it warrants more explanation (either through a question next lecture, office hours, or external resources). The exams are lengthier, cumulative problem sets. They are divided into sections which align more-or-less with the problem sets and lectures. Katz gives ample time to solve them because he gives them before breaks and expects them back afterward. However, one cannot work with classmates on these so while it is possible to learn the material for the exam in the time given, it is much more efficient to build up individual understanding throughout the semester.
External ResourcesAs mentioned before, Dummit and Foote's "Abstract Algebra" is the standard text for this course. While Katz teaches an unconventional version of Algebra I, it is still a fantastic and relevant text. Apart from that, I would look for PDFs online for materials with examples of topics covered in class. For example, while groups are defined in class, the course does not cover many examples of specific groups. Having a concept of a few major groups in mind can help one follow lectures and visualize problems more easily. Similarly, having some idea of Galois theory over polynomials with rational coefficients is helpful in understanding terms and theorems in the general theory.
What Students Should Know About This Course For Purposes Of Course SelectionMAT 345 is a challenging proof-based math class which is designed to teach the fundamentals of abstract algebra for higher level math courses. For math majors, it is a departmental requirement; the only consideration is whether or not one would like to take it with Professor Katz. Typically, Katz does not teach this course, so I would ask those who took it another year for information about the class when taught by other professors. For those considering the course out of interest, it is mainly for those who want to take more pure algebra courses at Princeton and beyond. Another course of interest is MAT 340 which focuses more on the applied aspect of groups in algebra.