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Helpful illustrations and exercises included throughout this lucid coverage of group theory, Galois theory and classical ideal theory stressing proof of important theorems. Includes many historical notes. Mathematical proof is emphasized. Includes 24 tables and figures. Reprint of the 1971 edition.
Chapter 1: Basic Terminology and Properties
* Summary:
* Defines abstract algebraic structures such as groups, rings, and fields.
* Introduces basic concepts like elements, operations, inverses, and identity elements.
* Explores properties like closure, associativity, and commutativity.
* Example:
* Consider the set of integers under addition. This forms a group where the elements are integers, the operation is addition, the identity element is 0, and every integer has an inverse (its negative).
Chapter 2: Groups
* Summary:
* Focuses on properties of groups, including order, subgroups, and cosets.
* Introduces Lagrange's Theorem, which relates the order of a group to the order of its subgroups.
* Explores concepts like cyclic groups and generators.
* Example:
* The set of complex numbers of the form e^(n*i*pi/3) for n = 0, 1, 2 forms a group under multiplication. It is cyclic, generated by e^(i*pi/3).
Chapter 3: Rings
* Summary:
* Defines rings and introduces concepts like ring homomorphisms, units, and ideals.
* Explores properties such as integral domains, fields, and quotient rings.
* Example:
* The set of all polynomials with real coefficients forms a ring under addition and multiplication. Its units are polynomials with non-zero constant terms.
Chapter 4: Euclidean Rings and Unique Factorization Domains
* Summary:
* Introduces Euclidean rings and unique factorization domains (UFDs).
* Explores properties such as prime and irreducible elements, as well as the Euclidean algorithm and unique factorization.
* Example:
* The ring of integers is a Euclidean ring, where the greatest common divisor (GCD) algorithm can be used to find the GCD of any two integers.
Chapter 5: Polynomials and Rational Expressions
* Summary:
* Focuses on polynomials and rational expressions over a field.
* Introduces concepts like polynomial rings, root finding, and factorization.
* Explores properties such as the Fundamental Theorem of Algebra, which states that every polynomial with complex coefficients can be factored into linear factors.
* Example:
* The polynomial x^3 - 2x^2 + 4x - 8 can be factored as (x - 2)(x - 4)(x + 2).
Chapter 6: Vector Spaces and Linear Transformations
* Summary:
* Introduces vector spaces and linear transformations.
* Explores concepts like basis, dimension, and vector subspaces.
* Discusses properties of linear transformations, including injectivity, surjectivity, and matrix representations.
* Example:
* The set of all vectors in R^3 forms a vector space. A linear transformation from R^3 to R^2 can be represented by a 2x3 matrix.
Chapter 7: Groups of Symmetries
* Summary:
* Focuses on finite groups and groups associated with symmetries.
* Introduces concepts like Cayley's Theorem, which states that every group can be represented as a group of permutations.
* Discusses the geometry of groups and their applications in crystallography and molecular symmetry.
* Example:
* The group of symmetries of a square is a dihedral group of order 8. It can be represented by the set of all permutations of the vertices of the square.
