# Research: Number Theory

## Summer School 2018: Algorithms for Extensions of Large Degree

From May 28 to June 1, 2018, the University of North Carolina at Greensboro will host the UNCG Summer School in Computational Number Theory: **Algorithms for Extensions of Large Degree**.

The summer school in computational number theory fills a gap in the education of many graduate students. Most graduate courses in number theory take a mainly theoretic approach with very little emphasis on the computational aspects of the subject. The goal of the UNCG Summer School in Computational Number Theory is to complement this with a constructive-algorithmic approach. Many of the algorithms used for number theoretic computations are non-trivial, which makes it difficult to cover them in a standard course.

Algorithms for extensions of global and local fields are the backbone of computational number theory. Improvements in computing power have made it feasible to conduct computations in larger and larger degree. As the complexity of algorithms for field extensions depends on the degree of the extensions, this has increased the interest in asymptotically fast algorithms. Among others we will consider algorithms for integral bases computation and ideal arithmetic.

On a typical day, external and local experts give talks in the morning, and in the afternoon students solve problems related to this material. The talks early in the week introduce the students to the subject. Talks later in the week 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 are aimed at increasing the students’ understanding of the material by working with it.

### Application

We expect to be able to provide support for approximately ten graduate students (travel, dorm accomodation, food). Interested students should complete the online application and have a letter of reference sent via email to batanged@uncg.edu by **March 22, 2018**. We will respond with funding decisions by April 13, 2018.

### Speakers

- Claus Fieker (Technische Universitä Kaiserslautern)
- Jordi Guardia (Universitat Politècnica de Catalunya)
- TBA

### Software

- +Ideals, a package for ideal arithmetic in number fields for Magma
- Nemo, a computer algebra package for the Julia programming language

### Organizers

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

### Acknowledgements

The summer school in computational number theory is supported by UNCG and the NSF (DMS-1602025).