Childs A Concrete Introduction to Higher Algebra

Erscheinungsjahr 2012
ISBN: 978-1-4684-0065-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 340 Seiten, eBook

Reihe: Undergraduate Texts in Mathematics

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The book developed out of a set of notes for a sophomore-junior level course at the State University of New York at Albany entitled Classical Algebra. In the 1950s and before, it was customary for the first course in algebra to be a course in the theory of equations, consisting of a study of polynomials over the complex, real, and rational numbers, and, to a lesser extent, linear algebra from the point of view of systems of equations. Abstract algebra, that is, the study of groups, rings, and fields, usually followed such a course. In recent years the theory of equations course has disappeared. Without it, students entering abstract algebra courses tend to lack the experience in the algebraic theory of the basic classical examples of the integers and polynomials necessary for understanding, and more importantly, for ap preciating the formalism. To meet this problem, several texts have recently appeared introducing algebra through number theory.

Zielgruppe

Lower undergraduate

Autoren/Hrsg.

Childs, Lindsay

Inhaltsverzeichnis

I INTEGERS.- 1 Numbers.- 2 Induction; the Binomial Theorem.- 3 Unique Factorization into Products of Primes.- 4 Primes.- 5 Bases.- 6 Congruences.- 7 Congruence Classes.- 8 Rings and Fields.- 9 Matrices and Vectors.- 10 Secret Codes, I.- 11 Fernjat’s Theorem, I: Abelian Groups.- 12 Repeating Decimals, I.- 13 Error Correcting Codes, I.- 14 The Chinese Remainder Theorem.- 15 Secret Codes, II.- II POLYNOMIALS.- 1 Polynomials.- 2 Unique Factorization.- 3 The Fundamental Theorem of Algebra.- 4 Irreducible Polynomials in ?[x].- 5 Partial Fractions.- 6 The Derivative of a Polynomial.- 7 Sturm’s Algorithm.- 8 Factoring in ?[x], I.- 9 Congruences Modulo a Polynomial.- 10 Fermat’s Theorem, II.- 11 Factoring in ?;[x], II: Lagrange Interpolation.- 12 Factoring in ?p[x].- 13 Factoring in ?[x], III: Mod m.- III FIELDS.- 1 Primitive Elements.- 2 Repeating Decimals, II.- 3 Testing for Primeness.- 4 Fourth Roots of One in ?p.- 5 Telephone Cable Splicing.- 6 Factoring in ?[x], IV: Bad Examples Modp.- 7 Congruence Classes Modulo a Polynomial: Simple Field Extensions.- 8 Polynomials and Roots.- 9 Error Correcting Codes, II.- 10 Isomorphisms, I.- 11 Finite Fields are Simple.- 12 Latin Squares.- 13 Irreducible Polynomials in ?p[x].- 14 Finite Fields.- 15 The Discriminant and Stickelberger’s Theorem.- 16 Quadratic Residues.- 17 Duplicate Bridge Tournaments.- 18 Algebraic Number Fields.- 19 Isomorphisms, II.- 20 Sums of Two Squares.- 21 On Unique Factorization.- Exercises Used in Subsequent Chapters.- Comments on the Starred Problems.- References.