# Algebraic Number Theory I

## Course on Algebraic Number Theory

Algebraic Number Theory is a fascinating area of mathematics, where the structure of the sets of natural numbers and of all integers, and natural generalizations of those, are studied by algebraic methods. While some of the underlying questions reach back several centuries, algebraic number theory is still an active and prospering area of research which has seen several spectacular new results and developments in the last few decades.

In the course, we will cover the basic theory of number fields, i.e., of the finite extension fields of $\mathbb Q$ (rings of integers, (prime) ideals, finiteness of the class group, …). Further information will follow later.

**Dates:** Tuesday 10-12 (S-U-3.02), Friday, 10-12 (S-U-3.01).

**Prerequisites:** Good knowledge of algebra (groups, rings and fields, field extensions, Galois theory).

**Exercise group:** There will be an exercise group lead by Dr. F. Fité, Friday, 8:30-10:00, S-U-3.02.

**Contact:** Ulrich Görtz, ulrich.goertz@uni-due.de

**Literature:**

There are many books on algebraic number theory. It might be best if you have a look at some of them yourself, in the library, to get an impression. A very incomplete list of books that are worth a look:

- E. Hecke, Lectures on the Theory of Algebraic Numbers (originally in German, a real classic)
- K. Ireland, M. Rosen, A classical introduction to modern number theory
- G. Janusz, Algebraic Number Fields
- D. Marcus, Number Fields
- J. Milne, Algebraic Number Theory
- J. Neukirch, Algebraische Zahlentheorie (also available in English)
- P. Samuel, Théorie algébrique des nombres (also available in English; very inexpensive)
- A. Schmidt, Einführung in die algebraische Zahlentheorie

## Problem Sheets

Due date | ||

Sheet 1 | Oct 27, 2015 | |

Sheet 2 | Nov 3, 2015 | |

Sheet 3 | Nov 10, 2015 | |

Sheet 4 | Nov 17, 2015 | |

Sheet 5 | Nov 24, 2015 | |

Sheet 6 | Dec 1, 2015 | |

Sheet 7 | Dec 8, 2015 | |

Sheet 8 | Dec 15, 2015 | |

Sheet 9 | Jan 12, 2016 | |

Sheet 10 | Jan 19, 2016 | |

Sheet 11 | Jan 26, 2016 | |

Sheet 12 | Feb 2, 2016 | |

Sheet 13 | Feb 9, 2016 | |

Preparation for the exam | – | |

Text of exam with solutions | – | |

Text of make up exam | – |