MTH 333: Topics in Abstract Algebra

Topic for Spring 2003: Algebraic Number theory

This course provides an introduction to the central problems and ideas of algebraic number theory. We extend what we mean by ``integers'' to include a larger class of numbers than Z= $\{0, \pm 1, \pm 2, \pm 3, \ldots \}$, and see that many properties of the ordinary integers ${\bf Z}$ generalize to these ``algebraic integers.'' This connection makes algebraic integers useful for solving problems that appear only to involve ordinary integers. The most famous example of this is Fermat's Last Theorem, which simply states that for $n > 2$ there are no positive integers $x$, $y$, and $z$ that satisfy $x^n+y^n=z^n$. In1850, Kummer used algebraic integers to prove Fermat's Last Theorem for a large set of prime exponents (although not all), and in doing so developed the field of algebraic number theory. It wasn't until the end of the 20th century, over three centuries after the death of Fermat, that Wiles finally proved Fermat's Last Theorem for all exponents $n$. One of the goals of this course is to understand the details of Kummer's proof, as well as a sketch of Wiles' difficult proof.

Topics from some recent years:

• (2001) Rings, Fields and Galois Theory.
  
• (2000) Rings, Fields and Galois Theory.
  
• (1999) Algebraic Number Theory.
  
• (1998) Linear Algebra: vector spaces, linear transformations, algebra of polynomials, canonical forms, inner product spaces.
  
• (1997) Computational Algebraic Geometry. Curves, surfaces, and higher-dimensional geometric configurations defined by polynomial equations.
  
• (1991) Algebraic Structures in Modern Geometry. Further study of groups, rings, and fields. Groups actions, fundamental groups, topics in algebraic geometry.
  
• (1990) Applications of Abstract Algebra. The theory of groups and rings applied to such topics as coding theory, statistical block design, automata, and computers.