有限場(chǎng)及伽羅瓦環(huán)講義LECTURES ON FINITE FIELDS AND GALOIS RINGS

內(nèi)容概要

This is a textbook for graduate and upper level undergraduate students in mathematics, computer science, communication engineering and other fields. The explicit construction of finite fields and the computation in finite fields are emphasised. In particular, the construction of irreducible polynomials and the normal basis of finite fields are included. The essentials of Galois rings are also presented. This invaluable book has been written in a friendly style, so that lecturers can easily use it as a text and students can use it for self-study. A great number of exercises have been incorporated. --This text refers to the Hardcover edition.

書籍目錄

1  Sets and Integers  1.1  Sets and Maps  1.2  The Factorization of Integers  1.3  Equivalence Relation and Partition  1.4  Exercises2  Groups  2.1  The Concept of a Group and Examples  2.2  Subgroups and Cosets  2.3  Cyclic Groups  2.4  Exercises3  Fields and Rings  3.1  Fields  3.2  The Characteristic of a Field    3.3  Rings and Integral Domains  3.4  Field of Fractions of an Integral Domain   3.5  Divisibility in a Ring  3.6  Exercises4  Polynomials  4.1  Polynomial Rings   4.2  Division Algorithm  4.3  Euclidean Algorithm  4.4  Unique Factorization of Polynomials  4.5  Exercises5  Residue Class Rings  5.1  Residue Class Rings  5.2  Examples  5.3  Residue Class Fields  5.4  More Examples  5.5  Exercises6  Structure of Finite Fields  6.1  The Multiplicative Group of a Finite Field  6.2  The Number of Elements in a Finite Field  6.3  Existence of Finite Field with pn Elements  6.4  Uniqueness of Finite Field with pn Elements  6.5  Subfields of Finite Fields  6.6  A Distinction between Finite Fields of Characteristic 2 and Not 2  6.7  Exercises7 Further Properties of Finite Fields  7.1  Automorphisms  7.2  Characteristic Polynomials and Minimal Polynomials  7.3  Primitive Polynomials  7.4  Trace and Norm  7.5  Quadratic Equations  7.6  Exercises8  Bases  8.1  Bases and Polynomial Bases  8.2  Dual Bases……9　Factoring Polynomials over Finite Fields 10　Irreducible Polynomials over Finite Fields 11　Quadratic Forms over Finite Fields 12　More Group Theory and Ring Theory 13　Hensel's Lemma and Hensel Lift 14　Galois Rings BibliographyIndex

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