Algebra and Number Theory: A Selection of Highlights (De Gruyter Textbook)

In the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines fundamental results in Algebra and Number Theory along with their proofs and their history. In the second edition, we include additional material on perfect and triangular numbers. We also added new sections on elementary Group Theory, p-adic numbers, and Galois Theory. A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications. Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested. Preface Contents 1 The Natural, Integral, and Rational Numbers 1.1 Number Theory and Axiomatic Systems 1.2 The Natural Numbers and Induction 1.3 The Integers Z 1.4 The Rational Numbers Q 1.5 The Absolute Value in N, Z and Q 2 Division and Factorization in the Integers 2.1 The Fundamental Theorem of Arithmetic 2.2 The Division Algorithm and the Greatest Common Divisor 2.3 The Euclidean Algorithm 2.4 Least Common Multiples 2.5 General Greatest Common Divisors and Lowest Common Divisors 3 Modular Arithmetic 3.1 The Ring of Integers Modulo n 3.2 Units and the Euler φ-Function 3.3 RSA Cryptosystem 3.4 The Chinese Remainder Theorem 3.5 Quadratic Residues 4 Exceptional Numbers 4.1 The Fibonacci Numbers 4.1.1 The Golden Rectangle 4.1.2 Squares in Semicircles 4.1.3 Side Length of a Regular 10-Gon 4.1.4 Construction of the Golden Section with Compass and Straightedge 4.2 Perfect Numbers and Mersenne Numbers 4.3 Fermat Numbers 5 Pythagorean Triples and Sums of Squares 5.1 The Pythagorean Theorem 5.2 Classification of the Pythagorean Triples 5.3 Sum of Squares 6 Polynomials and Unique Factorization 6.1 Polynomials over a Ring 6.2 Divisibility in Rings 6.3 The Ring of Polynomials over a Field 6.3.1 The Division Algorithm for Polynomials 6.3.2 Zeros of Polynomials 6.4 Horner-Scheme 6.5 The Euclidean Algorithm and Greatest Common Divisor of Polynomials over Fields 6.5.1 The Euclidean Algorithm for K[x] 6.5.2 Unique Factorization of Polynomials in K[x] 6.5.3 General Unique Factorization Domains 6.6 Polynomial Interpolation and the Shamir Secret Sharing Scheme 6.6.1 Secret Sharing 6.6.2 Polynomial Interpolation over a Field 6.6.3 The Shamir Secret Sharing Scheme 7 Field Extensions and Splitting Fields 7.1 Fields, Subfields and Characteristics 7.2 Field Extensions 7.3 Finite and Algebraic Field Extensions 7.3.1 Finite Fields 7.4 Splitting Fields 8 Permutations and Symmetric Polynomials 8.1 Permutations 8.2 Cycle Decomposition of a Permutation 8.2.1 Conjugate Elements in Sn 8.2.2 Marshall Hall's Theorem 8.3 Symmetric Polynomials 8.4 Some Topics in Group Theory 8.4.1 Cosets and Lagrange's Theorem 8.4.2 Normal Subgroups and Factor Groups 8.4.3 Group Isomorphism Theorems 8.4.4 Solvable Groups 8.4.5 Group Actions and the Sylow Theorems 8.4.6 The Fundamental Theorem of Finitely Generated Abelian Groups 9 Real Numbers 9.1 The Real Number System 9.2 Decimal Representation of Real Numbers 9.3 Periodic Decimal Numbers and the Rational Numbers 9.4 The Uncountability of R 9.5 Continued Fraction Representation of Real Numbers 9.6 Theorem of Dirichlet and Cauchy's Inequality 9.7 The p-adic Numbers 9.7.1 Normed Fields and Cauchy Completions 9.7.2 The p-adic Fields 9.7.3 The p-adic Norm 9.7.4 The Construction of Qp 9.7.5 Ostrowski's Theorem 9.7.6 The p-adic Arithmetic and p-adic Expansions 9.7.7 The p-adic Integers 9.7.8 Principal Ideals, Unique Factorization, and Completeness of Zp 9.7.9 Hensel's Lemma and Applications 9.7.10 The Non-Isomorphism of the p-adic Fields 10 The Complex Numbers, the Fundamental Theorem of Algebra, and Polynomial Equations 10.1 The Field C of Complex Numbers 10.2 The Complex Plane 10.2.1 Geometric Interpretation of Complex Operations 10.2.2 Polar Form and Euler's Identity 10.2.3 Other Constructions of C 10.2.4 The Gaussian Integers 10.3 The Fundamental Theorem of Algebra 10.3.1 First Proof of the Fundamental Theorem of Algebra 10.3.2 Second Proof of the Fundamental Theorem of Algebra 10.4 Solving Polynomial Equations in terms of Radicals 10.5 Galois Theory and the Solvability of Polynomial Equations in terms of Radicals 10.5.1 Automorphism Groups of Field Extensions 10.5.2 Finite Galois Extensions 10.5.3 The Fundamental Theorem of Galois Theory 10.5.4 Field Extensions by Radicals 10.5.5 Solvability by Radicals and Galois Extensions 10.6 Skew Field Extensions of C and Frobenius's Theorem 11 Quadratic Number Fields and Pell's Equation 11.1 Algebraic Extensions of Q 11.2 Algebraic and Transcendental Numbers 11.3 Discriminant and Norm 11.4 Algebraic Integers 11.4.1 The Ring of Algebraic Integers 11.5 Integral Bases 11.6 Quadratic Fields and Quadratic Integers 12 Transcendental Numbers and the Numbers e and π 12.1 The Numbers e and π 12.1.1 Calculation e of π 12.2 The Irrationality of e and π 12.3 The Numbers e and π throughout Mathematics 12.3.1 The Normal Distribution 12.3.2 The Gamma Function and Stirling's Approximation 12.3.3 The Wallis Product Formula 12.4 Existence of a Transcendental Number 12.5 The Transcendence of e and π 12.6 An Amazing Property of π and a Connection to Prime Numbers 13 Compass and Straightedge Constructions and the Classical Problems 13.1 Historical Remarks 13.2 Geometric Constructions 13.3 Four Classical Construction Problems 13.3.1 Squaring the Circle (Problem of Anaxagoras 500–428 BC) 13.3.2 The Doubling of the Cube or the Problem from Deli 13.3.3 The Trisection of an Angle 13.3.4 Construction of a Regular n-Gon 14 Euclidean Vector Spaces 14.1 Length and Angle 14.2 Orthogonality and Applications in R^2 and R^3 14.3 Orthonormalization and Closest Vector 14.4 Polynomial Approximation 14.5 Secret Sharing Scheme using the Closest Vector Theorem Bibliography Index A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications. Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested.(add bookmarks for all sections)