# Course on Algebraic Number Theory 2 (SS16)

## Algebraic Number Theory 2

This course is a continuation of the course Algebraic Number Theory 1 The main topic is the theory of the field of $p$-adic numbers and its finite extensions, and in particular *local class field theory* which gives a description of the maximal abelian quotient $\mathop{\rm Gal}(K^{\rm sep}/K)^{\rm ab}$ of such a field $K$ in terms of “$K$ itself”. Hopefully at the end there will be enough time left to sketch the implications of this for number fields.

**Prerequisites:** Linear Algebra and Algebra. Of course it is an advantage if you attended the Algebraic Number Theory 1 course, but this is not absolutely required. You do need some basic knowledge about fields with non-archimedean valuations however. Feel free to contact me if you need assistance with learning these things on your own.

**Dates:** Tue, Fri, 10-12 (First lecture: April 12). Place: S-U-3.02.

**Exercise group:** (Dr. F. Fité) Wed, 12-2pm, S-3.14.

## Problem sheets

Due | ||

Sheet 1 | April 26, 2016 | |

Sheet 2 | May 3, 2016 | |

Sheet 3 | May 10, 2016 | |

Sheet 4 | May 17, 2016 | |

Sheet 5 | May 24, 2016 | |

Sheet 6 | May 31, 2016 | |

Sheet 7 | June 7, 2016 | |

Sheet 8 | June 14, 2016 | |

Sheet 9 | June 21, 2016 | |

Sheet 10 | June 28, 2016 | |

Sheet 11 | July 5, 2016 | |

Sheet 12 | July 12, 2016 |

## Content of the course

Here is a rough outline of what we will cover:

- Hensel’s Lemma and consequences. Unramified, tamely ramfied, wildly ramified extensions of local fields.
- (if necessary) Infinite Galois theory
- Group Cohomology
- Local Class Field Theory
- Formal groups, Lubin-Tate theory
- (hopefully) Statement of the main results of Global Class Field Theory

**Literature:**

For large parts of the course, I will follow Milne’s notes:

- J. Milne, Class field theory

Further references that might be useful are

- Cassels, Fröhlich (eds.), Algebraic Number Theory
- Neukirch, Klassenkörpertheorie /Class field theory
- Neukirch, Zahlentheorie/Algebraic Number theory
- Artin, Tate, Class field theory
- T. Yoshida, Local class field theory via Lubin-Tate theory, Annales de la Faculte des Sciences de Toulouse, Ser. 6, 17-2 (2008), 411-438. math.NT/0606108
- Serre, Local fields / Corps locaux
- Fesenko, Vostokov, Local fields and their extensions, AMS Transl. of Math. Monographs 121

Survey articles on class field theory and further developments:

- R. Taylor, Reciprocity laws and density theorems
- K. Conrad, History of class field theory
- D. Cox, Primes of the form $x^2+ny^2$, Wiley (not a survey article, but a book which proves certain elementary (but difficult to prove!) statements about primes using global class field theory)