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
