Galois Theory (2012, M24) Notes

Algebraic and Algorithmic Aspects of Difference Equations∗

In this course, I will give an elementary introduction to the Galois theory of linear difference equations. This theory shows how to associate a group of matrices with a linear difference equation and shows how group theory can be used to determine properties of the solutions of the equations. I will begin by giving an introduction to the theory of linear algebraic groups, those groups that occ...

p-adic Galois Representations Math 162b Winter 2012 Lecture Notes

1 Local Class Field Theory 2 1.1 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Newton polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Ramification of local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Main results of local class field ...

A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...

Notes on Galois Theory

Math 806 Notes on Galois Theory

3 Extensions of rings and fields 14 3.1 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Integral ring extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Prime ideals in Z[x]: elementary classification . . . . . . . . . . . . . . . . . . . . . . 19 3.4 The spectrum of a commutative ring . . . . . . . . . . . . . ...