# 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 you 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 √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.

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.
In the past, Princeton introduced students to algebra through two courses. One of these (MAT 323), was a basic course covering groups, rings, and fields. The other (MAT 322), focused more on rings and fields, and also covered Galois theory. However, Princeton has restructured its introductory algebra courses. Now, there will be a two-semester algebra sequence. Algebra I will mainly focus on groups, and will also cover some representation theory. Algebra II will cover ring theory, field theory, and Galois theory. No further information is available at this point, but please check the math department’s undergraduate page for announcements. [/showhide]

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 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.
What to take before? Introductory Sequence.
What to take after? Class Field Theory, Local Fields.
MAT 454: Class Field Theory
Class Field Theory is about understanding the abelian extensions of a number field. The majority of the course is spent proving one very important theorem. The proof has two main parts – one of which uses analytical methods (from MAT 335), and the other of which uses group cohomology (you should at least be comfortable with exact sequences beforehand), which is more algebraic. MAT 419 leads into it quite nicely. Because its focus is on proving one big result, this course tends to have less emphasis on problem sets. However, you should plan on spending a lot of time reviewing your notes, as the material is quite challenging.
What to take before? Algebraic Number Theory.
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 416: Introduction to Algebraic Geometry (Kollar)
This is a course on varieties, which are sets of solutions to polynomial equations. Commutative algebra is a prerequisite, either in the form of MAT 447 or by reading Atiyah and MacDonald’s classic text and doing lots of exercises to get comfortable with the tools used in algebraic geometry. The course follows Shafarevich’s text and focuses on aspects of varieties, their local and global geometry, embeddings into projective space, and the specific case of curves which is extremely well-understood. The final third of the course consisted of student presentations about various special topics like elliptic curves, surfaces, resolutions of singularities, algebraic groups and others. This course is fast-paced and challenging, but worth the effort. Homeworks tended to vary in length, frequency and difficulty.
What to take before? Commutative Algebra (not absolutely necessary)
MAT 416: Introduction to Algebraic Geometry (Katz)
This is a course on sheaves, schemes and the cohomology of coherent sheaves on projective varieties. It follows the well-known text by Hartshorne. Commutative algebra is an absolute prerequisite and an introduction to varieties is highly recommended since schemes and sheaves are very abstract objects and having a good stock of examples in hand is vital to understand the material well. One unusual feature of this course which adds to its difficulty is that the bulk of the material on schemes and sheaves is relegated to the readings and homework while the instructor lectures on the cohomology of projective varieties. Thus in some sense, it is two courses rolled into one and one would be wise to treat it as such. This is arguably the most challenging course offered by the mathematics department due to the constantly steep learning curve and the exceptionally heavy workload. A final expository project makes for a fun finish to what certainly will be a gruelling semester.
What to take before? Commutative Algebra (a must).
What to take after? You’ll be ready for anything.
MAT 424: Representation Theory (Bhargava/Zhang)
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 tablesof 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.
MAT 424: Local Fields (Katz)
This course focuses on the Galois theory of the p-adic numbers, which are the canonical example of locally compact topological field with respect to a non-discrete topology. It’s very important to know Galois theory very well before taking this course, from MAT 346: Algebra II. After introducing the key techniques and ideas of the p-adics, the instructor devotes the rest of th semester to discussing various special topics — a variety of solutions to the inverse Galois problem for S_n and A_n, Hilbert’s irreducibility theorem and others. The homeworks are long but not difficult and are very good at getting students comfortable with the style of argument. There were no exams.
What to take before? Introductory Sequence
What to take after? Algebraic Number Theory, Class Field Theory.

Contacts [Show]Contacts [Hide]
Max Rabinovich ’13 (mrabinov[at]princeton[dot]edu) for general questions
Ashwath Rabindranath ’13 (arabindr[at]princeton[dot]edu) for algebraic number theory
Sarah Trebat Leder ’13 (strebat[at]princeton[dot]edu) for algebraic number theory