Advanced number theory

Includes index

Includes bibliography (p. 243-246) and references.

Background material : Review of elementary number theory and group theory : Number theoretic concepts : Congruence ; Unique factorization ; The Chinese remainder theorem ; Structure of reduced residue classes ; Residue classes for prime powers. Group theoretic concepts : Abelian groups and subgroups ; Decomposition into cyclic groups. Quadratic congruences : Quadratic residues ; Jacobi symbol
Characters : Definitions ; Total number of characters ; Residue classes ; Resolution modulus ; Quadratic residue characters ; Kronecker's symbol and Hasse's congruence ; Dirichlet's lemma on real characters
Some algebraic concepts : Representation by quadratic forms ; Use of surds ; Modules ; Quadratic integers ; Hilbert's example ; Fields ; Basis of quadratic integers ; Integral domain ; Basis of σn ; Fields of arbitrary degree
Basis theorems : Introduction of n dimensions ; Dirichlet's boxing-in principle ; Lattices ; Graphic representation ; Theorem on existence of basis ; Other interpretations of the basis construction ; Lattices of rational integers, canonical basis ; Sublattices and index concept ; Application to modules of quadratic integers ; Discriminant of a quadratic field ; Fields of higher degree
Further applications of basis theorems : Structure of finite abelian groups : Lattice of group relations ; Need for diagonal basis ; Elementary divisor theory ; Basis theorem for abelian groups ; Simplification of result. Geometric remarks on quadratic forms : Successive minima ; Binary forms ; Korkine and Zolatareff's example
Ideal theory in quadratic fields : Unique factorization and units : The "missing" factors ; Indecomposable integers, units, and primes ; Existence of units in a quadratic field ; Fundamental units ; Construction of a fundamental unit ; Failure of unique factorization into indecomposable integers ; Euclidean algorithm ; Occurrence of the Euclidean algorithm ; Pell's equation ; Fields of higher degree
Unique factorization into ideals : Set theoretical notation ; Definition of ideals ; Principal ideals ; Sum of ideals, basis ; Rules for transforming the ideal basis ; Product of ideals, the critical theorem, cancellation ; "To contain is to divide" ; Unique factorization ; Sum and product of factored ideals ; Two element basis, prime ideals ; The critical theorem and Hurwitz's lemma
Norms and ideal classes : Multiplicative property of norms ; Class structure ; Minkowski's theorem ; Norm estimate
Class structure in quadratic fields : The residue character theorem ; Primary numbers ; Determination of principal ideals with given norms ; Determination of equivalence classes ; Some imaginary fields ; Class number unity ; Units and class calculation of real quadratic fields ; The famous polynomials x2 + x + q
Applications of ideal theory : Class number formulas and primes in arithmetic progression : Introduction of analysis into number theory ; Lattice points in ellipse ; Ideal density in complex fields ; Ideal density in real fields ; Infinite series, the zeta-function ; Euler factorization ; The zeta-function and L-series for a field ; Connection with ideal classes ; Some simple class numbers ; Dirichlet L-series and primes in arithmetic progression ; Behavior of the L-series, conclusion of proof ; Weber's theorem on primes in ideal classes
Quadratic reciprocity : Rational use of class numbers ; Results on units ; Results on class structure ; Quadratic reciprocity preliminaries ; The main theorem ; Kronecker's symbol reappraised
Quadratic forms and ideals : The problem of distinguishing between conjugates ; The ordered bases of an ideal ; Strictly equivalent ideals ; Equivalence classes of quadratic forms ; The correspondence procedure ; The correspondence theorem ; Complete set of classes of quadratic forms ; Some typical representation problems
Compositions, orders, and genera : Composition of forms ; Orders, ideals, and forms ; Genus theory of forms ; Hilbert's description of genera

Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples, and more concrete, specific theorems than are found in most contemporary treatments of the subject.