| Preface |
| Preliminaries / Chapter 1.： |
| Notation and terminology / 1.： |
| Polynomial algebras / 2.： |
| Integral extensions / 3.： |
| Tensor products / 4.： |
| Module-theoretic prerequisites / 5.： |
| Topological prerequisites / 6.： |
| Classical Topics in Field Theory / Chapter 2.： |
| Algebraic extensions |
| Normal extensions |
| Separable, purely inseparable and simple extensions |
| Galois extensions |
| Finite fields, roots of unity and cyclotomic extensions |
| Norms, traces and their applications |
| Discriminants and integral bases / 7.： |
| Units in quadratic fields / 8.： |
| Units in pure cubic fields / 9.： |
| Finite Galois theory / 10.： |
| Profinite groups / 11.： |
| Infinite Galois theory / 12.： |
| Witt vectors / 13.： |
| Cyclic extensions / 14.： |
| Kummer theory / 15.： |
| Radical extensions and related results / 16.： |
| Degrees of sums in a separable field extension / 17.： |
| Galois cohomology / 18.： |
| The Brauer group of a field / 19.： |
| An interpretation of H[superscript 3 subscript O](G,E*) / 20.： |
| A cogalois theory for radical extensions / 21.： |
| Abelian p-extensions over fields of characteristic p / 22.： |
| Formally real fields / 23.： |
| Transcendental extensions / 24.： |
| Valuation Theory / Chapter 3.： |
| Valuations |
| Valuation rings and places |
| Dedekind domains |
| Completion of a field |
| Extensions of valuations |
| Valuations of algebraic number fields |
| Ramification index and residue degree |
| Structure of complete discrete valued fields |
| The equal characteristic case / A.： |
| The unequal characteristic case / C.： |
| The inertia field / D.： |
| Cyclotomic extensions of p-adic fields / E.： |
| Multiplicative Groups of Fields / Chapter 4.： |
| Some general observations |
| Infinite abelian groups |
| The Dirichlet-Chevalley-Hasse Unit Theorem |
| The torsion subgroup |
| Global fields |
| Algebraically closed, real closed and the rational p-adic fields |
| Local fields |
| Preparatory results |
| Extensions of algebraic number fields |
| Brandis's theorem |
| Fields with free multiplicative groups modulo torsion |
| A nonsplitting example |
| Embedding groups |
| Multiplicative groups under field extensions |
| Notes |
| Bibliography |
| Notation |
| Index |
| Preface |
| Preliminaries / Chapter 1.： |
| Notation and terminology / 1.： |
