Author(s): Neal H. McCoy
Publisher: Allyn and Bacon
Year: 1972
Language: English
City: Boston
Title
Contents
Preface
1. Some fundamental concepts
1.1. Sets
1.2. Mappings
1.3. Products of mappings
1.4. Equivalence relations
1.5. Operations
2. Rings
2.1. Formal properties of the integers
2.2. Definition of a ring
2.3. Examples of rings
2.4. Some properties of addition
2.5. Some other properties of a ring
2.6. General sums and products
2.7. Homomorphisms and isomorphisms
3. Integral domains
3.1. Definition of an integral domain
3.2. Ordered integral domains
3.3. Well-ordering and mathematical induction
3.4. A characterization of the ring of integers
3.5. The Peano axioms (optional)
4. Some properties of the integers
4.1. Divisors and the division algorithm
4.2. Different bases (optional)
4.3. Greatest common divisor
4.4. The fundamental theorem
4.5. Some applications of the fundamental theorem
4.6. Pythagorean triples (optional)
4.7. The ring of integers modulo n
5. Fields and the rational numbers
5.1. Fields
5.2. The characteristic
5.3. Some familiar notation
5.4. The field of rational numbers
5.5. A few properties of the field of rational numbers
5.6. Subfields and extensions
5.7. Construction of the integers from the natural numbers (optional)
6. Real and complex numbers
6.1. The field of real numbers
6.2. Some properties of the field of real numbers
6.3. The field of complex numbers
6.4. The conjugate of a complex number
6.5. Geometric representation and trigonometric form
6.6. The nth roots of a complex number
7. Groups
7.1. Definition and simple properties
7.2. Groups of permutations
7.3. Homomorphisms and isomorphisms
7.4. Cyclic groups
7.5. Cosets and Lagrange's theorem
7.6. The symmetric group S_n
7.7. Normal subgroups and quotient groups
7.8. Homomorphisms and subgroups
8. Finite abelian groups
8.1. Direct sums of subgroups
8.2. Cyclic subgroups and bases
8.3. Finite abelian p-groups
8.4. The principal theorems for finite abelian groups
9. The Sylow theorems
9.1. Conjugate theorems and transforms
9.2. Conjugate subgroups
9.3. Double cosets
9.4. Proofs of the Sylow theorems
10. Polynomials
10.1. Polynomial rings
10.2. The substitution process
10.3. Divisors and the division algorithm
10.4. Greatest common divisor
10.5. Unique factorization in F[x]
10.6. Rational roots of a polynomial over the rational field
10.7. Prime polynomials over the rational field (optional)
10.8. Polynomials over the real or complex numbers
10.9. Partial fractions (optional)
11. Ideals and quotient rings
11.1. Ideals
11.2. Quotient rings
11.3. Quotient rings F[x] / (s(x))
11.4. The fundamental theorem on ring homomorphisms
12. Vector spaces
12.1. Vectors in a plane
12.2. Definition and simple properties of a vector space
12.3. Linear dependence
12.4. Linear combinations and subspaces
12.5. Basis and dimension
12.6. Homomorphisms of vector spaces
12.7. Hom_F (V, W) as a vector space
12.8. Dual vector spaces
12.9. Quotient vector spaces and direct sums
12.10. Inner products in V_n (F)
13. Field extensions
13.1. The process of adjunction
13.2. The existence of certain extensions
13.3. Classifications of extensions
13.4. Simple extensions
13.5. Finite algebraic extensions
13.6. Equivalence of splitting fields of a polynomial
14. Systems of linear equations
14.1. Notation and simple results
14.2. Echelon systems
14.3. Matrices
14.4. Applications to systems of linear equations
14.5. Systems of linear homogeneous equations
15. Determinants
15.1. Preliminary remarks
15.2. General definition of determinant
15.3. Some fundamental properties
15.4. Expansion in terms of a row or column
15.5. The determinant rank of a matrix
15.6. Systems of linear equations
16. Linear transformations and matrices
16.1. Notation and preliminary remarks
16.2. Algebra of linear transformations
16.3. The finite dimensional case
16.4. Algebra of matrices
16.5. Linear transformations of V_n (F)
16.6. Adjoint and inverse of a matrix
16.7. Equivalence of matrices
16.8. The determinant of a product
16.9. Similarity of matrices
16.10. Invariant subspaces
16.11. Polynomials in a linear transformation
16.12. Characteristic vectors and characteristic roots
17. Some additional topics
17.1. Quaternions
17.2. Principal ideal domains
17.3. Modules
17.4. Modules over a principal ideal domain
17.5. Zorn's lemma
17.6. Representations of Boolean rings
Bibliography
Index
