ALGEBRA: FIELDS WITH STRUCTURE, ALGEBRAS, AND ADVANCED TOPICS

By K. MIZIM

CONTENTS ORDERED FIELDS AND REAL FIELDS ORDERED AND PREORDERED ?ELDS EXTENSIONS OF ?ELD ORDERS REAL-CLOSED ?ELDS THE FUNDAMENTAL THEOREM OF ALGEBRA ARTIN’S CHARACTERIZATION OF REAL-CLOSED ?ELDS SYLVESTER’S THEOREM ON THE NUMBER OF REAL ROOTS EXTENSION OF ORDER-PRESERVING HOMOMORPHISMS EXISTENCE OF REAL SPECIALIZATIONS HILBERT’S SEVENTEENTH PROBLEM AND THE REAL NULLSTELLENSATZ ARTIN’S SOLUTION TO HILBERT’S SEVENTEENTH PROBLEM GENERALIZATION TO AF?NE K-VARIETIES THE REAL NULLSTELLENSATZ POSITIVE DE?NITE FUNCTIONS ON SEMIALGEBRAIC SETS POSITIVE DE?NITE SYMMETRIC FUNCTIONS ORDERS AND QUADRATIC FORMS WITT EQUIVALENCE; THE WITT RING W K/ AND ITS PRIME IDEALS THE SPECTRUM OF W K/ THE TORSION ELEMENTS OF W K/ THE ZERO DIVISORS OF W K/ ABSOLUTE VALUES ON FIELDS ABSOLUTE VALUES ON THE RATIONALS NONARCHIMEDEAN ABSOLUTE VALUES COMPLETION OF ABSOLUTE VALUES THE ?ELD P OF P-ADIC NUMBERS EQUIVALENCE OF NORMS HENSEL’S LEMMA EXTENSION OF ABSOLUTE VALUES RESIDUE CLASS DEGREE AND RAMI?CATION INDEX DISCRETE ABSOLUTE VALUES THE FORMULA EF D N FOR COMPLETE, DISCRETE VALUATIONS UNRAMI?ED EXTENSIONS PURELY RAMI?ED EXTENSIONS LOCAL FIELDS CLASSI?CATION OF LOCAL ?ELDS CONNECTION WITH GLOBAL ?ELDS COMPLETION OF THE ALGEBRAIC CLOSURE OF P SOLVABILITY OF GALOIS GROUPS STRUCTURE OF THE MULTIPLICATIVE GROUP THE CASE OF P WITT VECTORS TEICHMÜLLER REPRESENTATIVES GHOST COMPONENTS WITT’S LEMMA THE RING OF WITT VECTORS HIGHER ARTIN–SCHREIER THEORY THE TSEN RANK OF A FIELD TSEN’S THEOREMS BEHAVIOR OF THE TSEN RANK WITH RESPECT TO EXTENSIONS NORM FORMS CI-?ELDS THE LANG–NAGATA THEOREM FINITE ?ELDS HAVE TSEN RANK 1 THE CHEVALLEY–WARNING THEOREM ALGEBRAICALLY CLOSED ?ELDS HAVE TSEN RANK 0 KRULL’S DIMENSION THEOREM FUNDAMENTALS OF MODULES FUNDAMENTALS OF LINEAR ALGEBRA SIMPLE AND SEMISIMPLE MODULES NOETHERIAN AND ARTINIAN MODULES THE JORDAN–HÖLDER THEOREM THE KRULL–REMAK–SCHMIDT THEOREM THE JACOBSON RADICAL NILPOTENCE OF THE RADICAL IN ARTINIAN RINGS ARTINIAN ALGEBRAS ARE NOETHERIAN WEDDERBURN THEORY SIMPLE ALGEBRAS DECOMPOSITION OF SIMPLE ALGEBRAS WEDDERBURN’S STRUCTURE THEOREM TENSOR PRODUCTS OF SIMPLE ALGEBRAS THE BRAUER GROUP OF A ?ELD TENSOR PRODUCTS OF SEMISIMPLE ALGEBRAS THE CENTRALIZER THEOREM AND SPLITTING ?ELDS THE SKOLEM–NOETHER THEOREM REDUCED NORM AND TRACE CROSSED PRODUCTS THE RELATIVE BRAUER GROUP OF GALOIS EXTENSIONS IN?ATION AND RESTRICTION THE BRAUER GROUP IS A TORSION GROUP CYCLIC ALGEBRAS QUATERNION ALGEBRAS COHOMOLOGY GROUPS AND THE CONNECTING HOMOMORPHISM CORESTRICTION THE BRAUER GROUP OF A LOCAL FIELD EXISTENCE OF UNRAMI?ED SPLITTING ?ELDS THE EQUALITY E D F D N FOR LOCAL DIVISION ALGEBRAS THE RELATIVE BRAUER GROUP IN THE UNRAMI?ED CASE THE HASSE INVARIANT CONSEQUENCES LOCAL CLASS FIELD THEORY THE LOCAL NORM RESIDUE SYMBOL FUNCTORIAL PROPERTIES OF THE NORM RESIDUE SYMBOL THE LOCAL RECIPROCITY LAW THE GROUP OF UNIVERSAL NORMS IS TRIVIAL THE LOCAL EXISTENCE THEOREM THE LOCAL KRONECKER–WEBER THEOREM SEMISIMPLE REPRESENTATIONS OF FINITE GROUPS TERMINOLOGY MASCHKE’S THEOREM APPLICATIONS OF WEDDERBURN THEORY ORTHOGONALITY RELATIONS FOR CHARACTERS INTEGRALITY PROPERTIES OF CHARACTERS INDUCED REPRESENTATIONS AND INDUCED CHARACTERS ARTIN’S INDUCTION THEOREM THE BRAUER–WITT INDUCTION THEOREM THE SCHUR GROUP OF A FIELD SCHUR INDEX OF ABSOLUTELY IRREDUCIBLE CHARACTERS SCHUR ALGEBRAS REDUCTION TO CYCLOTOMIC ALGEBRAS THE SCHUR GROUP OF A LOCAL ?ELD