Features

Exploratory Galois Theory adopts an exploration-based approach, using a variety of problems with hints and proof sketches to help students participate in the development of the theory.

The notion of algebraic number is developed from first principles, with explicit examples of algebraic numbers and finite extensions of the rationals (number fields) providing the primary context for a discussion of field extensions. In this setting of subfields of the complex numbers, some proofs and sketches may be successfully left for students to complete in exercises. Later on, in an optional section, the text outlines the changes necessary to expand the treatment to cover the Galois theory of finite extensions of arbitrary fields.

EGT encourages experimentation, with broadly functional Maple and Mathematica packages allowing students to examine extensions of the rationals generated by one or more algebraic numbers.  These packages, called AlgFields, permit students to use the powerful symbolic computation systems in a user-friendly way.

Extensions may be constructed using particular roots of irreducible polynomials, and Galois groups may be explicitly calculated. The functions employ the same procedures described in the text, such as factoring polynomials over extensions of Q, determining the irreducible factor corresponding to a particular root, determining whether a particular isomorphism may extend to a simple extension by determining what roots of a related polynomial lie in the target field. Free for educational distribution, the source code for the packages is available for interested students and faculty.

EGT assumes only a first course in abstract algebra (using any popular text, such as Gallian or Hungerford), with a review of necessary material in the first chapter.

EGT introduces concepts, theorems, and proofs at an extremely accessible level. The course may then be followed in one of several forms: a traditional lecture format, a seminar-style format with students presenting sections from the text, or a self-paced independent study.