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 charf
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
購物車/收銀台(
0
) | 在線留言板
| 付款方式
| 聯絡我們
| 運費計算
| 幫助中心 |
加入書簽
HOME
新書上架
