First Course in Algebra and Number Theory

Table of Contents 

I. Elementary Number Theory 
1-1. DIVISIBILITY 2 
1-2. THE DIVISION ALGORITHM 6 
1-3. THE GREATEST COMMON DIVISOR 13 
1-4. UNIQUE FACTORIZATION 21 
1-5. A CONVENIENT NOTATION 28 
1-6. LINEAR DIOPHANTINE EQUATIONS 42 
1-7. CONGRUENCE 50 
1-8. RADIX REPRESENTATION 64 
MISCELLANEOUS PROBLEMS 83

II. Rings and Domains 
2-1. RINGS: ELEMENTARY PROPERTIES 91 
2-2. EXAMPLES 104 
2-3. ORDERED AND WELL-ORDERED DOMAINS 138 
2-4. COMPUTATION RULES 160 
2-5. CHARACTERIZATION OF THE INTEGERS 174 
MISCELLANEOUS PROBLEMS 199

III. Congruences and Polynomials 
3-1. LINEAR CONGRUENCES 
3-2. UNITS AND FIELDS 
3-3. POLYNOMIALS AND POLYNOMIAL FUNCTIONS 
3-4. FACTORIZATION IN F[x] 
3-5. ROOTS OF POLYNOMIALS 
3-6. SOL YING POLYNOMIALS IN Zm[X] 
3-7. QUADRATIC RECIPROCITY 
MISCELLANEOUS PROBLEMS

IV. Groups 
BASIC FACTS AND EXAMPLES 
SUBGROUPS AND COSETS 
CYCLIC GROUPS 
NORMAL SUBGROUPS; FACTOR GROUPS; HOMOMORPHISMS 
PERMUTATION GROUPS 
THE GROUP Zm 
MISCELLANEOUS PROBLEMS

Selected Answers and Comment

Subject Index