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, 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.

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