Bolker E.D. Elementary Number Theory: An Algebraic Approach

New York: W.A. Benjamin, Inc., 1970. — xii, 180 p. — ISBN 0-8053-1018-5.This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and the Fermat conjecture for exponents three and four. The text contains abundant numerical examples and a particularly helpful collection of exercises, many of which are small research problems requiring substantial study or outside reading. Some problems call for new proofs for theorems already covered or for inductive explorations and proofs of theorems found in later chapters.Preface
Linear Diophantine Equations
Sums of Squares
Divisibility and Unique Factorization
The Diophantine Equation ax + b = c
The Diophantine Equation a₁x₁ + · · · + aₙxₙ = c
The Infinitude of the Primes
ProblemsCongruence
Arithmetic in ℤₙ: Solving Congruences
The Chinese Remainder Theorem
The Euler φ-Function
More about φ(n)
ProblemsPolynomials
The Algebra of Polynomials
Wilson's Theorem
The Diophantine Equation x² + y² = p
ProblemsThe Group of Units of ℤₙ
Decimal Expansions
Cyclic Groups
The Group Φ(p)
The Group Φ(2ᵅ)
The Group Φ(pᵅ)
The Group Φ(n)
ProblemsQuadratic Reciprocity
Residues
The Lemma of Gauss
The Quadratic Reciprocity Law
The Diophantine Equation x² ̶ my² = ±p
ProblemsQuadratic Number Fields
The Field ℚ(√m)
Algebraic Integers
Norms and Units
Units in Real Fields: Pell's Equation
Primes
Euclidean Number Fields
Consequences of Unique Factorization
The Diophantine Equation x² + 2y² = n
The Sum of Two Squares
A( ̶ 3) and Related Diophantine Equations
The Diophantine Equation x² ̶ 2y² = n
Quadratic Forms and Quadratic Number Fields
Sums of Four Squares
ProblemsThe Fermat Conjecture
Pythagorean Triples
x⁴ + y⁴ = z⁴; the Method of Descent
The Diophantine Equation x³ + y³ = z³
Cyclotomic Fields
ProblemsAppendicies
Unique Factorization
Primitive Roots
Indices for Φ(315)
Fundamental Units in Real Quadratic Number Fields
Chronological TableBibliography
List of Symbols
IndexФайл: отскан. стр. (b/w 600 dpi) + OCR + закладки