# Research: Number Theory

## Summer School 2013: Computational Algebraic Number Theory

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From May 20 to May 24, 2013, the University of North Carolina at Greensboro is hosting a summer school entitled **Computational Algebraic Number Theory**. This summer school will cover fundamental algorithms for number fields. It will complement what students learn in a standard course on algebraic number theory. The methods presented, such as methods for arithmetic, integral basis, unit groups, class group, and Galois group computations, are essential for further computations in number theory and algebraic geometry.

On a typical day, external and local experts will give talks in the morning, and in the afternoon students will solve problems related to this material. The talks early in the week will introduce the students to the subject. Talks later in the week will cover related areas of current research and unsolved problems. The problems given to the students might be of a theoretical nature but could also involve programming problems and computer experiments. All problems will be aimed at increasing the students’ understanding of the material by working with it.

### Schedule

All talks will take place in Room 213 in the Petty Building (**29** on the campus map). See directions for additional maps and directions.

Time | Monday 5/20 | Tuesday 5/21 | Wednesday 5/22 | Thursday 5/23 | Friday 5/24 |

9:00 | Welcome | Coffee | Coffee | Coffee | Coffee |

9:30 | Participants: Introductions I |
David Ford: Integral Bases notes handout slides exercises |
David Ford: Polynomial Factorization I notes |
David Ford: Polynomial Factorization II notes |
Dave Roberts: Computing Galois Groups II slides |

10:30 | Coffee | Coffee | Coffee | Coffee | Coffee |

10:45 | Participants: Introductions II slides |
Michael Pohst: Lattices slides exercises |
Michael Pohst: Unit and Class Groups I slides exercises |
Michael Pohst: Unit and Class Groups II slides exercises |
Michael Pohst: Diophantine equations slides exercises |

11:45 | Coffee | Coffee | Coffee | Coffee | Coffee |

12:00 | Dave Roberts: Introduction to Number Fields slides exercises exercisesmagma |
John Jones: Computing Number Fields I slides handout exercises |
Dave Roberts: Computing Galois Groups I slides |
John Jones: Computing Number Fields II |
John Jones: Computing Number Fields III |

13:00 | Lunch | Lunch | Lunch | Lunch | Lunch |

14:30 | Sebastian Pauli: Intro to Computing slides magma handbook |
Problem Session | Excursion | Problem Session | Problem Session |

15:00 | Problem Session | Problem Session | Problem Session | Problem Session | |

19:00 | Grasshoppers |

#### Workshop

Time | Saturday 5/25 |

9:00 | Coffee |

9:30 | Brian Sinclair: A Guide to the OM Algorithm |

10:30 | David Ford: Key Polynomials slides maclane v zx maclane v p |

11:30 | David Roberts: Hurwitz Number Fields slides |

12:30 | Michael Bush: Finding p-class towers of length 3 slides |

13:00 | Lunch |

### Participants

- Chad Awtrey (Elon University)
- Rebecca Black (University of Maryland)
- Michael Bush (Washington and Lee University)
- Thomas Alden Gassert (UMass Amherst)
- Bobby Grizzard (University of Texas at Austin)
- Anna Haensch (Wesleyan University)
- Jacob Hicks (University of Georgia)
- Avi Kulkarni (Simon Fraser University)
- Jonah Leshin (Brown University)
- Adam Lizzi (University of Maryland)
- Christine McMeekin (Cornell University)
- Khoa Nguyen (UC Berkeley)
- Caroline Turnage-Butterbaugh (University of Mississippi)

**Locals**

- Abraham Abebe (UNCG)
- Paul Duvall (UNCG)
- Ricky Farr (UNCG)
- Paula Hamby (UNCG)
- Jonathan Milstead (UNCG)
- Brian Sinclair (UNCG)

**Speakers**

- David Ford (Concordia University, Montreal)
- John Jones (Arizona State University)
- Michael Pohst (TU Berlin)
- David Roberts (University of Minnesota Morris)

**Organizers**

- Sebastian Pauli (UNCG)
- Brett Tangedal (UNCG)
- Dan Yasaki (UNCG)

### Local information

Information for visitors, including directions to UNCG and Petty Building.

### Acknowledgements

The summer school in computational number theory is supported by UNCG, the NSA (H98230-13-1-0253), and the NSF (DMS-1303565).