# Number Theory

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, MAT 346) and complex analysis (MAT335).These are essential tools in analytic number theory and algebraic number theory. Analytic number theory 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 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. 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.

Courses

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 later courses. This class is sometimes taken instead of MAT 215 as an introduction to proofs.

MAT 419: Arithmetic of Elliptic Curves
This course, which in Fall 2018 loosely followed Knapp’s text, begins with the study of elliptic curves over the rationals but quickly moves into elliptic functions and elliptic curves over the complex numbers, then into modular forms and L-functions; it culminated in an outline of the proof of Fermat’s Last Theorem. The problem sets were not unreasonable, but the content definitely ramped up – it is a very good idea to have a solid baseline understanding of analysis, algebra, elementary number theory and even topology in order to get the most out of the course.
Offered in the Fall.
What to take before? Complex Analysis.
What to take after?

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? Algebra I, Algebra II.
What to take after? Class Field Theory, Local Fields.

MAT 447: Commutative Algebra

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.

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.

Contacts

Ashwath Rabindranath ’13 (arabindr)
Sarah Trebat-Leder ’13 (strebat)

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