E-Book Overview
This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.
Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic
Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups. Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons. Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms. Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry. Commutative Rings III: Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases.
For all readers interested in abstract algebra.
E-Book Content
Special Notation
Set Theory and Number Theory
n r
bxc 8d (x) φ(n)
a|b (a, b) [a, b] a ≡ b mod m X ⊆Y X Y
X ×Y 1X |X | im f f : a 7→ b a≡b [a] [a]
m
natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 binomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .