Mathematical background

In addition to general mathematical maturity, this book assumes you have the following background:

• Basics of finite group theory
• Commutative rings, ideals, quotient rings
• Some elementary number theory
• Basic Galois theory of fields
• Point set topology
• Basic of topological rings, groups, and measure theory

For example, if you have never worked with finite groups before, you should read another book first. If you haven't seen much elementary ring theory, there is still hope, but you will have to do some additional reading and exercises. We will briefly review the basics of the Galois theory of number fields.

Some of the homework problems involve using a computer, but there are examples which you can build on. We will not assume that you have a programming background or know much about algorithms. Most of the book uses Sage http://sagemath.org, which is free open source mathematical software. The following is an example Sage session:

sage: 2 + 2
4 
sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1