| preface notes to the reader list of symbols i galois theory 1 field extensions 2 automorphisms 3 normal extensions 4 separable and inseparable extensions 5 the fundamental theorem of galois theory ii some galois extensions 6 finite fields 7 cyclotomic extensions 8 norms and traces 9 cyclic extensions 10 hilbert theorem 90 and group cohomology 11 kummer extensions iii applications of galois theory 12 discriminants 13 polynomials of degree 3 and 4 14 the transcendence of and e .15 ruler and compass constructions 16 solvability by radicals iv infinite algebraic extensions 17 infinite galois extensions 18 some infinite galois extensions v transcendental extensions 19 transcendence bases 20 linear disjointness 21 algebraic varieties 22 algebraic function fields 23 derivations and differentials appendix a ring theory 1 prime and maximal ideals 2 unique factorization domains 3 polynomials over a field 4 factorization in polynomial rings 5 irreducibility tests appendix b set theory 1 zorn's lemma 2 cardinality and cardinal arithmetic appendix c group theory 1 fundamentals of finite groups 2 the sylow theorems 3 solvable groups 4 profinite groups appendix d vector spaces 1 bases and dimension 2 linear transformations 3 systems of linear equations and determinants 4 tensor products appendix e topology 1 topological spaces 2 topological properties references index |
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