Numbers and Symmetry

This textbook presents modern algebra from the ground up using numbers and symmetry. Designed for a typical one-semester undergraduate course in modern algebra, it provides a gentle introduction to the subject by allowing students to see the ideas at work in accessible examples, rather than plunging them immediately into a sea of formalism. The student is involved at once with interesting algebraic structures, such as the Gaussian integers and the various rings of integers modulon, and is encouraged to take the time to explore and become familiar with those structures.

An undergraduate textbook that presents concrete examples before formal explanations and abstract definitions. Explains the basics of numeric groups, rings, and fields and the structures they form, and uses symmetry to provide visualizations of the central algebraic notion of isomorphism, similarities in form shown by different structures.
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This straightforward textbook teaches modern algebra from the ground up using numbers and symmetry as the basis of its approach. It explains the basics of numeric groups, rings and fields. It then examines the structures these groups form. Symmetry, the study of structure preserving transformations, provides the student with a visual introduction to the central algebraic notion of isomorphisms (similarities in form shown by different structures).

-- Explains basic concepts of modern algebra in a way that students can easily understand

-- Each chapter includes a section of notes devoted to explanatory, speculative, and historical asides

-- A full explanation of mathematical induction is included in an appendix

-- Allows students to become familiar with mathematical structures in concrete settings before facing abstract definitions

This textbook presents modern algebra from the ground up using numbers and symmetry. The idea of a ring and of a field are introduced in the context of concrete number systems. Groups arise from considering transformations of simple geometric objects. The analysis of symmetry provides the student with a visual introduction to the central algebraic notion of isomorphism.

Designed for a typical one-semester undergraduate course in modern algebra, it provides a gentle introduction to the subject by allowing students to see the ideas at work in accessible examples, rather than plunging them immediately into a sea of formalism. The student is involved at once with interesting algebraic structures, such as the Gaussian integers and the various rings of integers modulo n, and is encouraged to take the time to explore and become familiar with those structures.

In terms of classical algebraic structures, the text divides roughly into three parts: