域論

內(nèi)容概要

　　《域論(第2版)(英文版)》是一部研究生水平的域論的入門書籍。每節(jié)后面都有不少練習，使得本書既是一本很好的教程，也是一本不錯的參考書。本書從頭開始闡述了域基本理論，如果具備本科生水平的抽象代數(shù)知識將對學習本書具有很大的幫助。本書是第二版，作者基于第一版及在運用第一版在教學過程中的經(jīng)驗，又將本書中的基本內(nèi)容進行了改進。增加了新的練習和新的一章從歷史展望角度講述了
Galois理論，通書不斷涌現(xiàn)新話題，包括代數(shù)基本理論的證明、不可約情形的討論、Zp上多項式因式分解的Berlekamp代數(shù)等。目次：基礎；（第一部分）域擴展：多項式；域擴展；嵌入和可分性；代數(shù)獨立性；（第二部分）Galois理論Ⅰ，歷史回顧；Galois理論Ⅱ，理論；Galois理論Ⅲ，多項式的Galois群；域擴展作為向量空間；有限域Ⅰ，基本性質(zhì)；有限域Ⅱ，附加性質(zhì)；單位根；循環(huán)擴張；可解性擴張；（第三部分）二項式；二項式族。
　　

書籍目錄

preface
contents
0 preliminaries
 0.1 lattices
 0.2 groups
 0.3 the symmetric group
 0.4 rings
 0.5 integral domains
 0.6 unique factorization domains
 0.7 principal ideal domains
 0.8 euclidean domains
 0.9 tensor products
 exercises
part i-field extensions
1 polynomials
 1.1 polynomials over a ring
 1.2 primitive polynomials and irreducibility
 1.3 the division algorithm and its consequences
 1.4 splitting fields
 1.5 the minimal polynomial
 1.6 multiple roots
 1.7 testing for irreducibility
 exercises
2 field extensions
 2.1 the lattice of subfields of a field
 2.2 types of field extensions
 2.3 finitely generated extensions
 2.4 simple extensions
 2.5 finite extensions
 2.6 algebraic extensions
 2.7 algebraic closures
 2.8 embeddings and their extensions.
 2.9 splitting fields and normal extensions
 exercises
3 embeddings and separability
 3.1 recap and a useful lemma
 3.2 the number of extensions: separable degree
 3.3 separable extensions
 3.4 perfect fields
 3.5 pure inseparability
 3.6 separable and purely inseparable closures
 exercises
4 algebraic independence
 4.1 dependence relations
 4.2 algebraic dependence
 4.3 transcendence bases
 4.4 simple transcendental extensions
 exercises
part ii---galois theory
5 galois theory i: an historical perspective
 5.1 the quadratic equation
 5.2 the cubic and quartic equations
 5.3 higher-degree equations
 5.4 newton's contribution: symmetric polynomials
 5.5 vandermonde
 5.6 lagrange
 5.7 gauss
 5.8 back to lagrange
 5.9 galois
 5.10 a very brief look at the life of galois
6 galois theory i1: the theory
 6.1 galois connections
 6.2 the galois correspondence
 6.3 who's closed?
 6.4 normal subgroups and normal extensions
 6.5 more on galois groups
 6.6 abelian and cyclic extensions
 6.7 linear disjointness
 exercises
7 galois theory iii: the galois group of a polynomial
 7.1 the galois group of a polynomial
 7.2 symmetric polynomials
 7.3 the fundamental theorem of algebra.
 7.4 the discriminant of a polynomial
 7.5 the galois groups of some small-degree polynomials
 exercises
8 a field extension as a vector space
 8.1 the norm and the trace
 *8.2 characterizing bases
 *8.3 the normal basis theorem
 exercises
9 finite fields i: basic properties
 9.1 finite fields redux
 9.2 finite fields as splitting fields
 9.3 the subfields of a finite field.
 9.4 the multiplicative structure of a finite field
 9.5 the galois group of a finite field
 9.6 irreducible polynomials over finite fields
 *9.7 normal bases
 *9.8 the algebraic closure of a finite field
 exercises
10 finite fields i1: additional properties
 10.1 finite field arithmetic
 10.2 the number of irreducible polynomials
 10.3 polynomial functions
 10.4 linearized polynomials
 exercises
11 the roots of unity
 11.1 roots of unity
 11.2 cyclotomic extensions
 11.3 normal bases and roots of unity
 11.4 wedderburn's theorem
 11.5 realizing groups as galois groups
 exercises
12 cyclic extensions
 12.1 cyclic extensions
 12.2 extensions of degree char(f)
 exercises
13 solvable extensions
 13.1 solvable groups
 13.2 solvable extensions
 13.3 radical extensions
 13.4 solvability by radicals
 13.5 solvable equivalent to solvable by radicals
 13.6 natural and accessory irrationalities
 13.7 polynomial equations
 exercises
part iii--the theory of binomials
14 binomials
 14.1 irreducibility
 14.2 the galois group of a binomial
 14.3 the independence of irrational numbers
 exercises
15 families of binomials
 15.1 the splitting field
 15.2 dual groups and pairings
 15.3 kummer theory
 exercises
 appendix: mobius inversion
 partially ordered sets
 the incidence algebra of a partially ordered set
 classical mobius inversion
 multiplicative version of m6bius inversion
 references
 index

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