This course is an introduction to algebraic number theory (number fields, units, splitting of primes, class groups, etc.). In addition to theory, we will also discuss basic algorithms that allow (in principle) to explicitly compute most of the objects we will consider.

## Dates

Lectures are on **Mondays** and **Wednesdays**, both at **13:45–15:15** in room **48–438**. The first lecture is on Oct 28. Exercise sessions are on **Tuesdays**, **8:15–9:45** in room **48–438**.

## Exercise sheets

1 (28.10.), 2 (4.11.), 3 (11.11.), 4 (18.11.), 5 (25.11.), 6 (02.12.), 7 (09.12.), 8 (16.12), 9 (6.1.), 10 (13.1.), 11 (20.1.), 12 (3.2.)

## Additional material

- I turned Exercise 4.4(c) about the Hermite and Smith normal forms of a matrix over the Gaussian integers into a little computer algebra project. There, you'll also find the solution to the exercise.
- Here's the complete solution to Exercise 5.1(a), where we compute the maximal order in L(α) for α a root of X
^{3}-X^{2}-2X-8 using the round-2 algorithm. - Here's my (brutal) Magma program that computes units with the Dirichlet unit theorem (Exercise 9.3)
- Solution to Exercise 11.3.
- Solutions to Exercise Sheet 12.

## Lecture notes

# | Date | Content | Notes |
---|---|---|---|

1 | 28.10. | 1. What this is all about1.1. The sum of two squares problem 1.2. Review | |

2 | 30.10. | 2. Field extensions2.1 Constructing field extensions 2.2. Minimal polynomials and algebraic elements 2.3 Splitting fields | |

3 | 4.11. | 2.4 Conjugate roots and separability (continued) 2.5 Primitive elements 2.6 Characteristic polynomial, norm, trace 2.7 Trace form and discriminant | |

4 | 6.11. | 2.7 Trace form and discriminant (continued) 3. Rings of integers 3.1 Integral elements 3.2 Modules (review) 3.3 Integral elements form a ring | |

5 | 11.11. | 3.4 Ring of integers is integrally closed 3.5 Integrality of minimal polynomial, norm, trace 3.6 Ring of integers is finitely generated 3.7 Ring of integers is free (including review of free modules) | |

6 | 13.11. | 3.7 Ring of integers is free (continued, including structure of modules over PIDs) | |

7 | 18.11. | 3.7 Ring of integers is free (continued)4. Hermite and Smith normal form | |

8 | 20.11. | 4. Hermite and Smith normal form (continued)5. Finding an integral basis5.1 Orders, discriminants, a sufficient condition 5.2 Discriminant of an equation order 5.3 Zassenhaus approach | |

9 | 25.11. | 5.3 Zassenhaus approach (continued) 5.4 Review of prime ideals and radicals 5.5 Primes in an order 5.6 The round-2 algorithm (theory) | |

10 | 27.11. | 5.6 The round-2 algorithm (theory) (continued) 5.7 Computing in orders 5.8 Computing the p-radical | |

11 | 2.12. | 5.9 Computing the p-multiplier 5.10 Round-2 made constructive 6. Geometry of numbers (Minkowski theory)6.1 Lattices 6.2 Lattices in Euclidean space | |

12 | 4.12. | 6.3 Quadratic supplement and Cholesky decomposition 6.4 Minkowski theory | |

13 | 9.12. | 6.5 Discreteness of lattices 6.6 Shortest vectors and lattice density | |

14 | 11.12. | 6.6 Shortest vectors and lattice density (continued) 6.7 Successive minima 6.8 Lattice reduction | |

15 | 16.12. | 6.8 Lattice reduction (continued)7. Units7.1 Torsion units | |

16 | 18.12. | 7.2 Units are finitely generated 7.3 Free rank of the unit group | |

17 | 6.1. | 7.3 Free rank of the unit group (continued) 7.4 Remarks | |

18 | 8.1. | 8. Ideal theory of rings of integers8.1 Fractional ideals 8.2 Dedekind domains | |

19 | 13.1. | 8.2 Dedekind domains (continued) 8.3 Finiteness of the class group | |

20 | 15.1. | 8.3 Finiteness of the class group (continued) 8.4 Ramification theory | |

21 | 20.1. | 8.4 Ramification theory (continued) 8.5 Geometric interlude | |

22 | 22.1. | 8.5 Geometric interlude (continued) 8.6 Computing factorizations | |

23 | 27.1. | Lecture by Claus Fieker | |

24 | 29.1. | Lecture by Claus Fieker | |

25 | 3.2. | 8.7 (Un)ramified primes 8.8 Galois theory of primes | |

26 | 5.2. | 8.8 Galois theory of primes (continued) 8.9 Ramification in quadratic extensions | |

27 | 10.2. | 8.10 Ramification in cyclotomic fields 8.11 Quadratic reciprocity |

## Software

Computer algebra systems supporting number theory: Hecke (developed in Kaiserslautern!), PARI/GP, Sage, Magma. You can also try to develop algorithms from scratch in, e.g., Julia or Python.

## Literature

- J. Milne has excellent notes on
*Fields and Galois theory*and on*Algebraic Number Theory* - H. Cohen,
*A Course in Computational Algebraic Number Theory*(Springer GTM, 1996) - M. Pohst, H. Zassenhaus,
*Algorithmic Algebraic Number Theory*(Cambridge University Press, 1989) - J. Neukirch,
*Algebraic Number Theory*(Springer, 1999)

### Reading recommendations

- H.M. Edwards,
*The Background of Kummer's Proof of Fermat's Last Theorem for Regular Primes*(1975) - S. Gelbart,
*An elementary introduction to the Langlands program*(1984) - D. Wübben, D. Seethaler, J. Jaldén, and G. Matz,
*Lattice reduction—A survey with applications in wireless communications*(2011)