I am interested in a paper in the elective course Galois Theory, with a grade of 9-10. For this work, a good structure is needed, the text must be original, it must have at least as many references as are mentioned in the teacher’s detailed message that I have attached below. Make proper use of citations and be an overview article. The work can be either in Greek or in English, but it will be mainly appreciated if it is in English.
Detailed information:
What kind of project? I want you to write a survey article on the topic. pages long 15 (without the bibliography and the abstract), 17 with bibliography and abstract.
Which software should I use? LaTex Copyright. The text should be original, which means copy-paste is not allowed, beacause this would be an important copyright issue, which would imply failing the exam. There should be citations for all references you use in the text. Bibliography. At least 3 (non-greek) books and 2 articles. —- TOPIC: DIFFERENTIAL GALOIS THEORY Differential Galois theory studies the algebraic properties of solutions to differential equations. It extends classical Galois theory to differential equations.A central notion is that of a differential field, which is a field equipped with a derivation operator satisfying certain properties. Just as in classical Galois theory, for each differential equation, there is a corresponding differential Galois group, which captures the symmetries of its solutions.The differential Galois group is typically a matrix group over a differential field associated with the equation. The Picard-Vessiot theory provides a fundamental framework for understanding the relationship between differential fields and their differential Galois groups.It establishes a correspondence between the solutions of a linear differential equation and the structure of its differential Galois group. Differential Galois theory provides tools for determining whether a given differential equation is integrable in terms of elementary functions.This involves studying the differential Galois group and analyzing its properties, such as solvability and simplicity. Differential Galois theory has applications in various areas of mathematics and physics, including mathematical physics, algebraic geometry, and computer algebra systems.It plays a crucial role in the study of integrable systems, quantum mechanics, and differential equations arising in theoretical physics.
Some References `Galois Theory of Linear Differential Equations` by Marius van der Put and Michael F. Singer provides a comprehensive introduction to the subject. `Lectures on Differential Galois Theory` by A. Magid is a classic text that covers the basics of the theory. `Galois Theory of Linear Differential Equations` by Marius van der Put and Michael F. Singer focuses on the linear case and its connections to algebraic geometry. `Differential Algebra and Algebraic Groups` by E. Kolchin discusses the relationship between differential algebra and algebraic groups, which is foundational for differential Galois theory.
