Rings and Fields

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This accessible introduction to the mathematics of rings and fields shows how algebraic techniques can be used to solve many difficult problems. The book is carefully organized. Prerequisite mathematical skills are introduced at the beginning, and each chapter opens with an introduction to a problem followed by the development of the algebraic techniques necessary for its solution, using concrete mathematical and non-mathematical examples. Although prior knowledge of group theory is not required, a chapter is included which states the axiom for a group and proves the group theoretic results needed in Galois theory. The work is intended for advanced students and researchers in mathematics, computer science, and electronic engineering.

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Product Details

  • ISBN-13: 9780198534556
  • Publisher: Oxford University Press, USA
  • Publication date: 12/28/1992
  • Series: Oxford Science Publications
  • Pages: 184
  • Product dimensions: 9.50 (w) x 6.31 (h) x 0.63 (d)

Meet the Author

University College, Galway
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Table of Contents

0 Preliminaries 1
Definition of rings and fields
Vector spaces
Equivalence relations
Axiom of choice
1 Diophantine equations: Euclidean domains 13
Euclidean domain of Gaussian integers
Euclidean domains as unique factorization domains
2 Construction of projective planes: splitting fields and finite fields 25
Existence and uniqueness of splitting fields and of finite fields of prime power order
3 Error codes: primitive elements and subfields 49
Existence of primitive elements in finite fields
Subfields of finite fields
Computation of minimum polynomials
4 Construction of primitive polynomials: cyclotomic polynomials and factorization 65
Basic properties of cyclotomic polynomials
Berlekamp's factorization algorithm
5 Ruler and compass constructions: irreducibility and constructibility 83
Product formula for the degree of composite extensions
Irreducibility criteria for polynomials over the rationals
The field of constructible real numbers
6 Pappus' theorem and Desargues' theorem in projective planes: Wedderburn's theorem 93
Proof of Wedderburn's theorem
7 Solution of polynomials by radicals: Galois groups 109
Basic definitions and results in Galois groups
8 Introduction to groups 135
Group axioms
Subgroup lattice
Class equation
Cauchy's theorem
Transitive permutation groups
Soluble groups
9 Crytography: elliptic curves and factorization 151
Euler's function
Discrete logarithms
Elliptic curves
Pollard's method of factorizing integers
Elliptic curve factorization of integers
Further reading 166
Index 167
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