product iamge

Excursions in Number Theory, Algebra, and Analysis

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Download
Search on Amazon

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

This textbook originates from a course taught by the late Ken Ireland in 1972. Designed to explore the theoretical underpinnings of undergraduate mathematics, the course focused on interrelationships and hands-on experience. Readers of this textbook will be taken on a modern rendering of Ireland’s path of discovery, consisting of excursions into number theory, algebra, and analysis. Replete with surprising connections, deep insights, and brilliantly curated invitations to try problems at just the right moment, this journey weaves a rich body of knowledge that is ideal for those going on to study or teach mathematics.

A pool of 200 ‘Dialing In’ problems opens the book, providing fuel for active enquiry throughout a course. The following chapters develop theory to illuminate the observations and roadblocks encountered in the problems, situating them in the broader mathematical landscape. Topics cover polygons and modular arithmetic; the fundamental theorems of arithmetic and algebra; irrational, algebraic and transcendental numbers; and Fourier series and Gauss sums. A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Return trips to the Dialing In problems are encouraged, offering opportunities to put theory into practice and make lasting connections along the way.

Excursions in Number Theory, Algebra, and Analysis invites readers on a journey as important as the destination. Suitable for a senior capstone, professional development for practicing teachers, or independent reading, this textbook offers insights and skills valuable to math majors and high school teachers alike. A background in real analysis and abstract algebra is assumed, though the most important prerequisite is a willingness to put pen to paper and do some mathematics.

Author(s): Kenneth Ireland, Al Cuoco
Series: Undergraduate Texts in Mathematics: Readings in Mathematics
Publisher: Springer
Year: 2023

Language: English
Pages: 198
City: Cham

Preface
History
From Ken’s Original Preface
Using This Book
Acknowledgments
Recollections from Colleagues and Friends
Outline placeholder
From Ken Ribet
From the University of New Brunswick
Contents
Notation
1 Dialing In Problems
1.1 Dialing In Set 1
1.2 Dialing In Set 2
1.3 Dialing In Set 3
1.4 Dialing In Set 4
1.5 Dialing In Set 5
1.6 Dialing In Set 6
1.7 Dialing In Set 7
1.8 Dialing In Set 8
2 Polygons and Modular Arithmetic
2.1 The Complex Numbers
2.2 The Pentagon, Gauss, and Kronecker
2.2.1 A Theorem of Kronecker
2.2.2 Some Properties of Algebraic Extensions of Fields
2.2.3 Now We Can Show That mathbbQ[ζn] Is a Field
2.2.4 A Criterion for Irreducibility
2.3 Modular Arithmetic
2.3.1 Quadratic Reciprocity
2.4 Supplement: Dirichlet's Theorem on Primes in Arithmetic Progression
2.5 A Little Group Theory
2.6 Orbits and Elementary Group Theory
3 The Fundamental Theorem of Arithmetic
3.1 Getting Started
3.1.1 Computing Greatest Common Divisors
3.1.2 Modular Arithmetic with Polynomials
3.2 The Gaussian Integers
3.3 The Two Square Theorem
3.3.1 Fermat's Last Theorem
3.4 Formal Dirichlet Series and the Number of Representations of an Integer as the Sum of Two Squares
3.4.1 Formal Dirichlet Series
3.5 Supplement: Hilbert's 17th Problem
4 The Fundamental Theorem of Algebra
4.1 Getting Started
4.2 Background from Elementary Analysis
4.3 First Proof of the Fundamental Theorem of Algebra: An Analytic Approach
4.4 Background from the Theory of Equations
4.5 Second Proof of the Fundamental Theorem of Algebra: All Algebra (Almost)
4.5.1 The Idea behind the Proof of Theorem 4.11: More Modular Arithmetic with Polynomials
4.6 Galois Theory and the Fundamental Theorem of Algebra
4.7 The Topological Point of View
4.8 Supplement: and Its Factors
5 Irrational, Algebraic, and Transcendental Numbers
5.1 Liouville's Observation
5.2 Gelfond–Schneider and Lindemann–Weierstrass
5.3 The Irrationality of
5.4 The Irrationality of and
5.4.1 Next up:
5.5 The Transcendence of
5.6 Is Transcendental
5.6.1 More About Symmetric Functions
5.6.2 Euler's Identity
5.6.3 Setting the Stage
5.6.4 And Now … the Proof
6 Fourier Series and Gauss Sums
6.1 The Fourier Series of a Differentiable Function and
6.2 Dirichlet's Theorem
6.3 Applications to Numerical Series
6.4 Gauss Sums
6.4.1 A Brief Review of Infinite Integrals
6.4.2 Using Complex Numbers
6.4.3 The Value of the Gauss Sum
6.5 On and
Bibliography
Index