Number theory is a branch of mathematics devoted primarily to the study of the integers, their additive and multiplicative structures and their properties that set them apart from other rings (structures with addition and multiplication). Questions that pertain to the integers (and generalizations of them) are said to be arithmetic.
Elementary number theory refers to number theory which does not use tools from algebra and analysis. Princeton’s course on this is MAT 214. Many math majors who are interested in number theory learn this material on their own instead of taking the course.
In order to proceed further, one must first study algebra (MAT 345, 346) and complex analysis (MAT335).These are essential tools in analytic number theory and algebraic number theory. Analytic number theory (MAT 415) uses tools from analysis to study the integers, and often is concerned with questions about the asymptotics and distribution of arithmetic data, like prime numbers, class groups of number fields, discriminants of number fields, and so on. Complex analysis is especially important, and in fact much of the second half of MAT 335 is concerned with the proof of the prime number theorem, one of the pioneering efforts in analytic number theory.
Algebraic number theory (MAT 415, MAT 447) uses algebraic techniques to study number fields, which are finite field extensions of the rational numbers. Number fields have very similar properties to the rational numbers (they have a ring of integers that behaves much like Z) but differ from them in subtle ways that are of arithmetic interest. Unlike over the rational numbers, unique factorization of integers in a number field as the product of prime numbers does not always hold, but an analogous version for ideals does!
A third branch of number theory is arithmetic geometry, which is concerned with finding solutions of polynomial equations over fields that are not algebraically closed, such as the rational numbers. This subject is closely related to algebraic geometry, which is the study of geometric objects defined by polynomial equations. With the rapid development of algebraic geometry over the last fifty years, arithmetic geometry has become a very exciting field of study for today’s mathematicians. Algebraic geometry is usually offered as MAT 416, which can be either a course on varieties or schemes and cohomology, depending on the professor. Commutative algebra, taught in MAT 447, is a key tool in studying algebraic number theory and algebraic geometry, and it is a good idea to study it before doing too much algebraic geometry (the algebraic number theory course does not assume knowledge of it). Algebraic geometry is notoriously frightening, but is essential for research in the mathematics of today — so it’s best to learn it early!
Number theory uses a surprising amount of representation theory, topology, differential geometry, real analysis and combinatorics — in this field, more than any other, a broad base is crucial to be able to do research today.
A helpful link [Show]A helpful link [Hide]
Sam Ruth, a Princeton grad student, wrote a great article about learning algebraic number theory, which goes into a lot of depth about what you should learn as an undergrad and grad student if you want to be a number theorist, and talks about some of the main ideas and theorems in algebraic number theory, analytic number theory, and elliptic curves.
Here is the link.
Courses directly relevant to future number theorists [Show]Courses directly relevant to future number theorists [Hide]
MAT 214: Numbers, Equations, and Proofs
This is a class in elementary number theory. The material in this class is important to know if you want to continue in number theory, and will sometimes be assumed in MAT 419. This class is sometimes taken instead of MAT 215 as an introduction to proofs. Please see the math department’s page on this course for a description.
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.
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.
MAT 415: Analytic Number Theory
This course will be taught in Fall 2012 for the first time in a while, so no additional information is known about this course. Please check the math department’s page and the registrar’s page for more information.
MAT 447: Commutative Algebra
This course will be taught in Fall 2012 differently than in the past, so no additional information is known about this course. Please check the math department’s page and the registrar’s page for more information.
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.
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 grueling semester.
MAT 455: Advanced Topics in Geometry – Lie Theory
The goal of this course is to study the structure theory of Lie groups and Lie algebras. These objects are ubiquitous in mathematics and are studied using a variety of algebraic, analytic and geometric techniques. This course covers the geometry, structure theory, classification and touches upon their representation theories. Some background in differential geometry is essential, mostly material from the first few weeks of MAT 355. Alternatively, reading through the first few chapters of Spivak’s book on Differential Geometry should suffice. A variety of textbooks are useful — in particular, Adams, Humphreys and Bump.
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 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.
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 the 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.