# Algebra

Algebra is perhaps best defined as the study of “algebraic structures.” Broadly construed, algebraic structures are sets endowed with additional operations—think of the integers (a set) with addition and multiplication (additional operations). Usually, these operations allow elements to be combined in some way, but sometimes they’re better described as allowing transformation of elements into each other as, for instance, in the case of the category of sets endowed with the usual mappings between sets. Such structures are ubiquitous: integers and their various extensions, the vector spaces studied in linear algebra, groups of transformations that encode symmetries, abstract categories that capture relationships between complex objects (like groups themselves) are all examples. Due to this ubiquity, algebra crops up in all areas of mathematics and has a host of applications.

The first two structures you are likely to encounter are groups and rings. In general, groups should be thought of as collections of transformations. That’s because objects in a group can be composed (usually called “multiplication”) and inverted, just like functions, but nothing else is required for a set to have the structure of a group. Groups are important and beautiful objects. For instance, the set of all invertible linear transformations of a vector space to itself is a group, as is the set of all rotations of space (of any number of dimensions). These examples also illustrate the interpretation of groups as symmetries of some other mathematical entity–in this case, a vector space. Groups also play this role in the setting of permutations (symmetries of finite sets), geometric rearrangements of shapes (symmetries of polygons), and Galois theory (symmetries of number systems).

This last example shows how groups can be connected to their more sophisticated cousins, rings. While a ring can be thought of as a group with additional structure, a ring is really a very different animal. Rather than functions and transformations, rings represent numbers. In this case, you can add them, take their negatives, and multiply them—but that’s it. Since many natural number systems lack division (is one divided by two an integer?), it is excluded from the defining properties of rings. When a ring has it anyway, it is called a field–thus, we speak of the “ring of integers,” but the “field of rational (or real, or complex) numbers.” Rings serve as a natural generalization of the integers, and they provide a setting for the study both of abstract objects that resemble them and concrete number systems extending them, like the rational numbers, or the set of all numbers of the form $a + b\sqrt{5}$ with $a, b$ integers.

Now, all of this is just the beginning. We haven’t even mentioned the question of commutativity, or the fact that there is a natural way to think of rings as collections of functions (just ask an algebraic geometer). Nor have we really discussed the rich theory that results in studying the way algebraic objects interact with other mathematical entities. Nor the details of Galois theory. Nor categories. The list goes on and on. But Princeton offers many ways for you to get into these and other algebraic topics. This begins with an introductory sequence that starts with groups, develops the fundamental aspects of the theory behind them, then builds up elementary ring theory and the most important parts of classical Galois theory.

After you’ve completed the introductory sequence, you will be adequately prepared to jump into the more advanced courses that investigate the topics we’ve mentioned above.

In order to graduate with a mathematics degree, it is required to complete at least one algebra course.

Introductory courses

Princeton’s introductory algebra courses cover the three basic algebraic structures: groups, rings, and fields. A group is a structure with one binary operation, such as the set of permutations of {1, 2, …, n}, with the binary operation being composition. A ring is a structure with addition and multiplication, such as the set of polynomials with real coefficients. A field is a ring where each nonzero element has a multiplicative inverse, such as the set of real or complex numbers.

As of Fall 2016, the introductory courses are MAT 345 (mainly groups, rings, and maybe Galois theory or representation theory depending on the teacher) and MAT 346 (sometimes Galois theory; in Spring 2018/19, this has covered local field theory). Neither course has a consistent instructor or text. However, good references for relevant material include the Abstract Algebra text by Dummit and Foote, as well as Algebra by Michael Artin.

MAT 419: Algebraic Number Theory
This course uses a lot of algebra, so you should definitely have taken MAT 345 and MAT 346 first. Knowing Galois theory is especially important. A bit of complex analysis is used towards the end of the course, so having taken MAT 335 beforehand or concurrently would be helpful (there are even a few times where the same proof is done in both classes). There are problem sets approximately every week, which are time consuming but extremely worthwhile. The text for the class is by Samuel, but as it is a very thin book, it can often be helpful to look at other texts, such as Marcus. This course covers quite a lot of material, including some class field theory.
Offered in the Spring.
What to take before? Introductory Sequence.
What to take after? Class Field Theory, Local Fields.

MAT 447: Commutative Algebra
Please check registrar’s page or math department website for information on this course.
What to take before? Introductory Sequence.
What to take after? Introduction to Algebraic Geometry.

MAT 449: Representation Theory
Typically taught by Professor Morel, this course draws on many different areas of mathematics – algebra, geometry and some combinatorics. The first half of the course focuses on studying the linear representations of finite groups — this is a completely worked out theory and the course spent substantial time working out various examples of character tables of finite groups. The second half of the course focused on the representation theory of Lie algebras, in particular on the theory of highest weights. The difference between MAT 455 and this course is that the primary goal in this course is to classify the irreducible representations of the objects in question, while in MAT 455 the emphasis is on their intrinsic geometry — hence, there is value to be derived in taking both courses. The textbook followed is usually Fulton and Harris. In case you are new to working with tensor, symmetric and wedge products, spend some time before the course starts to become familiar with them. Of course, knowledge of basic group theory is essential.
What to take before? Introductory Sequence.
What to take after? You’ll be ready for anything.

Contacts

Max Rabinovich ’13 (mrabinov) for general questions
Ashwath Rabindranath ’13 (arabindr) for algebraic number theory
Sarah Trebat Leder ’13 (strebat) for algebraic number theory

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