Algebra, Number Theory and Combinatorics

Algebra is one of the fundamental areas of mathematics. Like most of modern mathematics, it is no exaggeration to say that Algebra is very abstract.  The many abstract structures and constructions that exist in Algebra can be difficult to grasp upon first encounter. For this reason, it is sometimes helpful to have a “handle” to lend support.  In its essence, Algebra is the study of polynomial equations.  While not intending to be an oversimplification of the matter, keeping this in mind can be of help to a student trying to make sense of the many abstract notions that arise.

For instance, Number Theory can be considered as that subset of Algebra that is concerned with polynomial equations for which the coefficients involve only natural numbers.  Likewise, the origins of Group Theory lie in the study of solutions to polynomial equations in one variable. It was Galois who stressed the importance of looking at the permutations of the set of roots of a polynomial in one indeterminate. This led to what is now called Galois Theory, as well as to the notion of a group acting on a set, hence to what is now called Group Theory.

The set of solutions to a system of polynomials in several variables is called an algebraic variety. Algebraic Geometry arose as the study of algebraic varieties. Linear Algebra is the study of systems of linear equations.  Arising out of this study are what we now call vector spaces, and more generally, modules.  Matrices turn out to have both practical and theoretical importance in Linear Algebra. Ring Theory can be thought of as the natural abstraction of the addition and multiplication operations possessed by the set of square matrices.  Commutative Algebra naturally developed out of the study of properties of rings of functions on algebraic varieties.

A polynomial equation in one indeterminate that has no repeated root is said to be separable. The recently published book “Separable Algebras” by Professor T. J. Ford, is concerned with algebraic structures that arise as abstractions of the notion of separable polynomials. Included, for example, are the study of Azumaya algebras, the henselization of local rings,and Galois theory. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders and class groups.