Advanced Number Theory

by Cohn, Harvey

• ISBN13: 9780486640235
• ISBN10: 048664023X
• Format: Paperback
• Copyright: 1980-08-01
• Publisher: Dover Publications

Summary

Eminent mathematician/teacher approaches algebraic number theory from historical standpoint. Demonstrates how concepts, definitions, theories have evolved during last 2 centuries. Abounds with numerical examples, over 200 problems, many concrete, specific theorems. Includes numerous graphs and tables.

Table of Contents

Introductory Survey	1	(1)
Diophantine Equations	1	(1)
Motivating Problem in Quadratic Forms	2	(3)
Use of Algebraic Numbers	5	(1)
Primes in Arithmetic Progression	6	(3)
PART 1. BACKGROUND MATERIAL
Review of Elementary Number Theory and Group Theory	9	(13)
Number Theoretic Concepts	9	(1)
Congruence	9	(1)
Unique factorization	10	(1)
The Chinese remainder theorem	11	(1)
Structure of reduced residue classes	12	(1)
Residue classes for prime powers	13	(2)
Group Theoretic Concepts	15	(1)
Abelian groups and subgroups	15	(1)
Decomposition into cyclic groups	16	(2)
Quadratic Congruences	18	(1)
Quadratic residues	18	(2)
Jacobi symbol	20	(2)
Characters	22	(17)
Definitions	22	(2)
Total number of characters	24	(3)
Residue classes	27	(1)
Resolution modulus	28	(4)
Quadratic residue characters	32	(3)
Kronecker's symbol and Hasse's congruence	35	(1)
Dirichlet's lemma on real characters	36	(3)
Some Algebraic Concepts	39	(15)
Representation by quadratic forms	39	(1)
Use of surds	40	(1)
Modules	41	(1)
Quadratic integers	42	(1)
Hilbert's example	43	(1)
Fields	44	(1)
Basis of quadratic integers	45	(2)
Integral domain	47	(1)
Basis of σn	48	(1)
Fields of arbitrary degree	49	(5)
Basis Theorems	54	(21)
Introduction of n dimensions	54	(1)
Dirichlet's boxing-in principle	54	(1)
Lattices	55	(2)
Graphic representation	57	(1)
Theorem on existence of basis	58	(5)
Other interpretations of the basis construction	63	(2)
Lattices of rational integers, canonical basis	65	(3)
Sublattices and index concept	68	(2)
Application to modules of quadratic integers	70	(2)
Discriminant of a quadratic field	72	(1)
Fields of higher degree	73	(2)
Further Applications of Basis Theorems	75	(18)
Structure of Finite Abelian Groups	75	(1)
Lattice of group relations	75	(1)
Need for diagonal basis	76	(1)
Elementary divisor theory	77	(4)
Basis theorem for abelian groups	81	(1)
Simplification of result	82	(1)
Geometric Remarks on Quadratic Forms	83	(1)
Successive minima	83	(3)
Binary forms	86	(2)
Korkine and Zolatareff's example	88	(5)
PART 2. IDEAL THEORY IN QUADRATIC FIELDS
Unique Factorization and Units	93	(20)
The ``missing'' factors	93	(1)
Indecomposable integers, units, and primes	94	(1)
Existence of units in a quadratic field	95	(3)
Fundamental units	98	(2)
Construction of a fundamental unit	100	(2)
Failure of unique factorization into indecomposable integers	102	(2)
Euclidean algorithm	104	(1)
Occurrence of the Euclidean algorithm	105	(5)
Pell's equation	110	(1)
Fields of higher degree	111	(2)
Unique Factorization into Ideals	113	(18)
Set theoretical notation	113	(1)
Definition of ideals	114	(2)
Principal ideals	116	(1)
Sum of ideals, basis	117	(2)
Rules for transforming the ideal basis	119	(1)
Product of ideals, the critical theorem, cancellation	120	(2)
``To contain is to divide''	122	(1)
Unique factorization	123	(1)
Sum and product of factored ideals	124	(1)
Two element basis, prime ideals	125	(3)
The critical theorem and Hurwitz's lemma	128	(3)
Norms and Ideal Classes	131	(11)
Multiplicative property of norms	131	(3)
Class structure	134	(2)
Minkowski's theorem	136	(3)
Norm estimate	139	(3)
Class Structure in Quadratic Fields	142	(17)
The residue character theorem	142	(3)
Primary numbers	145	(2)
Determination of principal ideals with given norms	147	(1)
Determination of equivalence classes	148	(1)
Some imaginary fields	149	(2)
Class number unity	151	(1)
Units and class calculation of real quadratic fields	151	(4)
The famous polynomials x2 + x + q	155	(4)
PART 3. APPLICATIONS OF IDEAL THEORY
Class Number Formulas and Primes in Arithmetic Progression	159	(24)
Introduction of analysis into number theory	159	(1)
Lattice points in ellipse	160	(1)
Ideal density in complex fields	161	(2)
Ideal density in real fields	163	(3)
Infinite series, the zeta-function	166	(1)
Euler factorization	167	(2)
The zeta-function and L-series for a field	169	(1)
Connection with ideal classes	170	(3)
Some simple class numbers	173	(1)
Dirichlet L-series and primes in arithmetic progression	174	(2)
Behavior of the L-series, conclusion of proof	176	(3)
Weber's theorem on primes in ideal classes	179	(4)
Quadratic Reciprocity	183	(12)
Rational use of class numbers	183	(1)
Results on units	184	(3)
Results on class structure	187	(3)
Quadratic reciprocity preliminaries	190	(2)
The main theorem	192	(1)
Kronecker's symbol reappraised	193	(2)
Quadratic Forms and Ideals	195	(17)
The problem of distinguishing between conjugates	195	(1)
The ordered bases of an ideal	196	(1)
Strictly equivalent ideals	197	(1)
Equivalence classes of quadratic forms	198	(2)
The correspondence procedure	200	(4)
The correspondence theorem	204	(3)
Complete set of classes of quadratic forms	207	(2)
Some typical representation problems	209	(3)
Compositions, Orders, and Genera	212	(19)
Composition of forms	212	(4)
Orders, ideals, and forms	216	(5)
Genus theory of forms	221	(7)
Hilbert's description of genera	228	(3)
CONCLUDING SURVEY	231	(12)
Cyclotomic Fields and Gaussian Sums	232	(2)
Class Fields	234	(4)
Global and Local Viewpoints	238	(5)
Bibliography and Comments	243	(4)
Some Classics Prior to 1900	243	(1)
Some Recent Books (After 1900)	244	(1)
Special References by Chapter	245	(2)
Appendix Tables	247	(28)
I. Minimum Prime Divisors of Numbers Not Divisible by 2, 3, or 5 from 1 to 18,000	247	(8)
II. Power Residues for Primes Less than 100	255	(6)
III. Class Structures of Quadratic Fields of √m for m Less than 100	261	(14)
Index	275