LARGE FIELDS IN DIFFERENTIAL GALOIS THEORY

Fields and Galois Theory

These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental extensions. They are an expansion of those handed out during a course taught to first-year graduate students. v2.01 (August 21, 1996). First version on the web. v2.02 (May 27, 1998). Fixed about 40 minor errors; 57 pages. v3.00 (April 3, 2...

Differential Galois Theory

The aim of the talk is to explain how to make these statements precise and prove them, using differential Galois theory. The reference to all of the material here is [vdPS03]. By solving a differential equation we mean obtaining a solution via a finite number of operations of the following kind (starting with a rational function): • Adding a function algebraic over the functions we already have...

Differential Galois Theory

Differential Galois Theory is a branch of abstract algebra that studies fields equipped with a derivation function. In much the same way as ordinary Galois Theory studies field extensions generated by solutions of polynomials over a base field, differential Galois Theory studies differential field extensions generated by solutions to differential equations over a base field. In this paper, we w...

Non-linear Differential Galois Theory

Galois Theory, Splitting Fields, and Computer Algebra

We provide some algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra. c © 2006 Elsevier Ltd. All rights reserved.