PROBLEMS IN ALGEBRAIC NUMBER THEORY

By WILLIAM SMITH

CONTENTS I PROBLEMS ELEMENTARY NUMBER THEORY INTEGERS APPLICATIONS OF UNIQUE FACTORIZATION THE ABC CONJECTURE SUPPLEMENTARY PROBLEMS EUCLIDEAN RINGS PRELIMINARIES GAUSSIAN INTEGERS EISENSTEIN INTEGERS SOME FURTHER EXAMPLES ALGEBRAIC NUMBERS AND INTEGERS BASIC CONCEPTS LIOUVILLE’S THEOREM AND GENERALIZATIONS ALGEBRAIC NUMBER FIELDS INTEGRAL BASES THE NORM AND THE TRACE EXISTENCE OF AN INTEGRAL BASIS EXAMPLES IDEALS IN OK DEDEKIND DOMAINS INTEGRAL CLOSURE CHARACTERIZING DEDEKIND DOMAINS FRACTIONAL IDEALS AND UNIQUE FACTORIZATION DEDEKIND’S THEOREM FACTORIZATION IN OK THE IDEAL CLASS GROUP ELEMENTARY RESULTS FINITENESS OF THE IDEAL CLASS GROUP DIOPHANTINE EQUATIONS EXPONENTS OF IDEAL CLASS GROUPS QUADRATIC RECIPROCITY GAUSS SUMS THE LAW OF QUADRATIC RECIPROCITY QUADRATIC FIELDS PRIMES IN SPECIAL PROGRESSIONS THE STRUCTURE OF UNITS DIRICHLET’S UNIT THEOREM UNITS IN REAL QUADRATIC FIELDS HIGHER RECIPROCITY LAWS CUBIC RECIPROCITY EISENSTEIN RECIPROCITY 10 ANALYTIC METHODS 10 1 THE RIEMANN AND DEDEKIND ZETA FUNCTIONS 10 2 ZETA FUNCTIONS OF QUADRATIC FIELDS 10 3 DIRICHLET’S L-FUNCTIONS 10 4 PRIMES IN ARITHMETIC PROGRESSIONS 10 5 SUPPLEMENTARY PROBLEMS 11 DENSITY THEOREMS 11 1 COUNTING IDEALS IN A FIXED IDEAL CLASS 11 2 DISTRIBUTION OF PRIME IDEALS 11 3 THE CHEBOTAREV DENSITY THEOREM 11 4 SUPPLEMENTARY PROBLEMS II SOLUTIONS