Description

If a = 3, b = 4, and c = 5, we have a² + b² = c² . A triple (a, b, c) of positive integers satisfying a² + b² = c² is called a Pythagorean triple. It is possible to determine all Pythagorean triples explicitly, and the first step in their determination is to factor the equation a² = c² − b² in the form    a² = (c + b)(c − b). On the other hand, it is not possible to find a triple (a, b, c) of positive integers satisfying aⁿ + bⁿ = cⁿ for an integer n > 2; this fact is known as Fermat’s last theorem. After being open for about 300 years, this theorem was proved in the 1990’s by the effort of several mathematicians, notably Andrew Wiles. An important first step in this solution, as realized by Ernst Eduard Kummer in the nineteenth century, is to factor the equation aⁿ = cⁿ − bⁿ in a way that is analogous to the above factorization of  a² = c² − b². This is no longer possible with integer numbers alone. Rather, one has to extend the rational numbers to so-called algebraic number fields, and study certain numbers in these fields that behave like the integers behave in the rational numbers. These numbers are therefore called algebraic integers. Algebraic number theory is the study of the properties of these generalized integers. Typical questions are whether these integers can also be factored into primes, whether this decomposition is unique, and whether an ordinary prime number in the integers remains prime when considered inside a larger algebraic number field.

Objectives

The objective of the course is to introduce the student to the foundations of algebraic number theory. Topics covered are integrality and integral bases, Dedekind rings, ideal class groups, the class number, the theorems of Minkowski and Hermite, Dirichlet’s unit theorem, quadratic number fields and cyclotomic fields, decomposition group and inertia group, and Hilbert’s ramification theory. Depending on the available time and the choice of the instructor, additional topics may include p-adic numbers, valuation theory, and local fields.

Texbooks

• Z. I. Borevich/I. R. Shafarevich: Number theory, Pure Appl. Math., Vol. 20,
  Academic Press, Orlando, 1966.
• G. J. Janusz: Algebraic number fields, 2nd ed., Grad. Stud. Math., Vol. 7, Am.
  Math. Soc., Providence, 1996.
• J. Neukirch: Algebraic number theory, Grundlehren Math. Wiss., Vol. 322,
  Springer, Berlin, 1999.