Articoli correlati a An Introduction to Commutative Algebra and Number Theory
Sinossi
An Introduction to Commutative Algebra and Number Theory is an elementary introduction to these subjects. Beginning with a concise review of groups, rings and fields, the author presents topics in algebra from a distinctly number-theoretic perspective and sprinkles number theory results throughout his presentation. The topics in algebra include polynomial rings, UFD, PID, and Euclidean domains; and field extensions, modules, and Dedekind domains.
In the section on number theory, in addition to covering elementary congruence results, the laws of quadratic reciprocity and basics of algebraic number fields, this book gives glimpses into some deeper aspects of the subject. These include Warning's and Chevally's theorems in the finite field sections, and many results of additive number theory, such as the derivation of LaGrange's four-square theorem from Minkowski's result in the geometry of numbers.
With addition of remarks and comments and with references in the bibliography, the author stimulates readers to explore the subject beyond the scope of this book.
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Contenuti
PRELIMINARIES: GROUPS, RINGS, AND FIELDS
Groups
Rings and Fields
THE INTEGERS
Divisibility
The Fundamental Theorem of Arithmetic
The Chinese Remainder Theorem
POLYNOMIAL RINGS
Rings of Polynomials
Division in Polynomial Rings
Exercise Set A
RINGS AND FIELDS REVISITED
Characteristic of a Ring
Wilson's Theorem
A Result on Vector Spaces
FACTORIZATION
Divisibility
UFD and PID
Euclidean Domains
GAUSS LEMMA AND EISENSTEIN CRITERION
Gauss Lemma
Eisenstein Criterion
FIELD EXTENSIONS
Algebraic Extensions
Normal Extensions
Separable Extensions
Finite Fields
QUADRATIC RECIPROCITY LAW
Exercise Set B
MODULES
Basic Definitions
A Result on Finitely Generated Modules
Noetherian Modules
Modules over a PID
Some Special Results
GAUSSIAN INTEGERS AND THE RING Z [v-5]
Gaussian Integers
The Ring Z [v-5]
ALGEBRAIC NUMBER FIELDS-I
Integral Dependence
Integers in Number Fields
Exercise Set C
DEDEKIND DOMAINS
Fractional Ideals
Properties of Dedekind Domains
ALGEBRAIC NUMBER FIELDS-II
Class Groups
Discriminants
Some Results in Geometry of Numbers
An Estimation
Dirichlet's Unit Theorem
Exercise Set D
QUADRATIC FIELDS
Integral Bases and Discriminants
Splitting of Rational Primes
The Group of Units
Norm-Euclidean Number Fields
Solutions to Selected Exercises
Appendix: Lucas-Lehmer Test
Bibliography
Index
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