| Foreword | p. v |
| Ordered Fields and Real Fields | p. 1 |
| Ordered and preordered fields | p. 2 |
| Extensions of field orders | p. 4 |
| Real-closed fields | p. 5 |
| The fundamental theorem of algebra | p. 7 |
| Artin's characterization of real-closed fields | p. 8 |
| Sylvester's theorem on the number of real roots | p. 10 |
| Extension of order-preserving homomorphisms | p. 11 |
| Existence of real specializations | p. 12 |
| Hilbert's Seventeenth Problem and the Real Nullstellensatz | p. 15 |
| Artin's solution to Hilbert's seventeenth problem | p. 15 |
| Generalization to affine K-varieties | p. 16 |
| The real Nullstellensatz | p. 18 |
| Positive definite functions on semialgebraic sets | p. 23 |
| Positive definite symmetric functions | p. 25 |
| Orders and Quadratic Forms | p. 29 |
| Witt equivalence; the Witt ring W (K) and its prime ideals | p. 30 |
| The spectrum of W (K) | p. 32 |
| The torsion elements of W (K) | p. 34 |
| The zero divisors of W (K) | p. 37 |
| Absolute Values on Fields | p. 39 |
| Absolute values on the rationals | p. 40 |
| Nonarchimedean absolute values | p. 45 |
| Completion of absolute values | p. 47 |
| The field Q[subscript p] of p-adic numbers | p. 52 |
| Equivalence of norms | p. 54 |
| Hensel's Lemma | p. 56 |
| Extension of absolute values | p. 59 |
| Residue Class Degree and Ramification Index | p. 65 |
| Discrete absolute values | p. 67 |
| The formula ef = n for complete, discrete valuations | p. 68 |
| Unramified extensions | p. 70 |
| Purely ramified extensions | p. 73 |
| Local Fields | p. 75 |
| Classification of local fields | p. 76 |
| Connection with global fields | p. 78 |
| Completion of the algebraic closure of Q[subscript p] | p. 80 |
| Solvability of Galois groups | p. 81 |
| Structure of the multiplicative group | p. 83 |
| The case of Q[subscript p] | p. 91 |
| Witt Vectors | p. 93 |
| Teichmuller representatives | p. 94 |
| Ghost components | p. 96 |
| Witt's Lemma | p. 98 |
| The ring of Witt vectors | p. 100 |
| Higher Artin-Schreier theory | p. 105 |
| The Tsen Rank of a Field | p. 109 |
| Tsen's theorems | p. 110 |
| Behavior of the Tsen rank with respect to extensions | p. 112 |
| Norm forms | p. 114 |
| C[subscript i]-fields | p. 116 |
| The Lang-Nagata Theorem | p. 118 |
| Finite fields have Tsen rank 1 | p. 121 |
| The Chevalley-Warning Theorem | p. 122 |
| Algebraically closed fields have Tsen rank 0 | p. 123 |
| Krull's dimension theorem | p. 124 |
| Fundamentals of Modules | p. 127 |
| Fundamentals of linear algebra | p. 128 |
| Simple and semisimple modules | p. 133 |
| Noetherian and artinian modules | p. 139 |
| The Jordan-Holder Theorem | p. 142 |
| The Krull-Remak-Schmidt Theorem | p. 143 |
| The Jacobson radical | p. 144 |
| Nilpotence of the radical in artinian rings | p. 148 |
| Artinian algebras are noetherian | p. 149 |
| Wedderburn Theory | p. 151 |
| Simple algebras | p. 151 |
| Decomposition of simple algebras | p. 152 |
| Wedderburn's structure theorem | p. 155 |
| Tensor products of simple algebras | p. 160 |
| The Brauer group of a field | p. 164 |
| Tensor products of semisimple algebras | p. 166 |
| The Centralizer Theorem and splitting fields | p. 168 |
| The Skolem-Noether Theorem | p. 173 |
| Reduced norm and trace | p. 176 |
| Crossed Products | p. 183 |
| The relative Brauer group of Galois extensions | p. 184 |
| Inflation and restriction | p. 190 |
| The Brauer group is a torsion group | p. 194 |
| Cyclic algebras | p. 196 |
| Quaternion algebras | p. 203 |
| Cohomology groups and the connecting homomorphism | p. 212 |
| Corestriction | p. 217 |
| The Brauer Group of a Local Field | p. 223 |
| Existence of unramified splitting fields | p. 224 |
| The equality e = f = n for local division algebras | p. 226 |
| The relative Brauer group in the unramified case | p. 229 |
| The Hasse invariant | p. 231 |
| Consequences | p. 234 |
| Local Class Field Theory | p. 239 |
| The local norm residue symbol | p. 240 |
| Functorial properties of the norm residue symbol | p. 242 |
| The local reciprocity law | p. 243 |
| The group of universal norms is trivial | p. 247 |
| The local existence theorem | p. 249 |
| The local Kronecker-Weber theorem | p. 251 |
| Semisimple Representations of Finite Groups | p. 253 |
| Terminology | p. 254 |
| Maschke's Theorem | p. 259 |
| Applications of Wedderburn theory | p. 261 |
| Orthogonality relations for characters | p. 265 |
| Integrality properties of characters | p. 268 |
| Induced representations and induced characters | p. 272 |
| Artin's induction theorem | p. 275 |
| The Brauer-Witt induction theorem | p. 277 |
| The Schur Group of a Field | p. 283 |
| Schur index of absolutely irreducible characters | p. 284 |
| Schur algebras | p. 286 |
| Reduction to cyclotomic algebras | p. 288 |
| The Schur group of a local field | p. 294 |
| Problems and Remarks | p. 299 |
| Index | p. 331 |
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