Topics in this book

These are some of the main topics that are discussed in this book:

• Rings of integers of number fields
• Unique factorization of ideals in Dedekind domains
• Structure of the group of units of the ring of integers
• Finiteness of the group of equivalence classes of ideals of the ring of integers (the ``class group'')
• Decomposition and inertia groups, Frobenius elements
• Ramification
• Discriminant and different
• Quadratic and biquadratic fields
• Cyclotomic fields (and applications)
• How to use a computer to compute with many of the above objects (both algorithms and actual use of PARI and ).
• Valuations on fields
• Completions ($p$-adic fields)
• Adeles and Ideles

Note that we will not do anything nontrivial with zeta functions or $L$-functions. This is to keep the prerequisites to algebra, and so we will have more time to discuss algorithmic questions. Depending on time and your inclination, I may also talk about integer factorization, primality testing, or complex multiplication elliptic curves (which are closely related to quadratic imaginary fields).