# Field theory

There have been 1 completed talk, 1 document, and 2 topic suggestions tagged with **field theory**.

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## Completed Talks

### Inverse Galois Theory

Delivered by David Liu on Friday March 17, 2017

Almost 200 years after Évariste Galois's death, there is still one question about Galois groups — the symmetries of the roots of polynomials, that still remains unsolved. This is the Inverse Galois Problem — whether every group is a Galois group of a Galois extension of the rational numbers. In this talk, I will give an overview of the progress that has been made, the approaches that mathematicians are making, and directions for further research.

A note on the requirements for this talk: It is recommended that you be comfortable with groups, fields, field extensions, automorphisms, and basic Galois theory. This requirement can be met by any of the following suggested alternatives:

having taken PMATH 347, and currently taking PMATH 348;

having taken PMATH 347, and coming for the prerequisite knowledge presentation (see below);

having equivalent knowledge to PMATH 347 and PMATH 348;

otherwise, if you are currently taking or have taken MATH 146, this talk is still accessible, but it is highly recommended that you read An Introduction to Galois Theory by Dan Goodman, and watch this 18-minute lecture by Matthew Salomone (after reading the article) and optionally also come for the prerequisite knowledge presentation

A prerequisite knowledge presentation about Galois theory will be given at 17:00, prior to the beginning of the talk at 17:30. This presentation will last about 20 minutes. If you are unfamiliar with the material, it is recommended that you read some of the linked material above in addition to attending this presentation. Attending this presentation is optional. The reference material used for this presentation is the Galois Theory document.

## Documents

### Galois Theory

This is a reference document on Galois Theory by Akshay Tiwary and Fengyang Wang. It covers a subset of the material of PMATH 348.

## Talk Suggestions

### Costas Arrays

Permutation matrices for which all $\frac{n(n-1)}{2}$ displacements between all pairs of 1s are distinct are known as Costas arrays, and these have important applications in sonar. The Welch construction of Costas arrays uses concepts from number theory, whereas the Lempel-Golomb construction uses concepts from field theory.

Possible reference materials for this topic include

Quick links: Google search, arXiv.org search, propose to present a talk

algebra field theory number theory prime number

### Galois Field Arithmetic

A Galois field is a finite field and are used in a variety of applications, including in classical coding theory and cryptography algorithms. This topic studies how to efficiently optimize arithmetic in such fields.

Possible reference materials for this topic include

Quick links: Google search, arXiv.org search, propose to present a talk

algebra algorithm computer science cryptography efficiency field theory