FIELDS AND GALOIS THEORY

By J. GILAA

CONTENTS 1 RINGS AND FIELDS DE?NITIONS AND BASIC PROPERTIES SUBRINGS, IDEALS AND HOMOMORPHISMS THE FIELD OF FRACTIONS OF AN INTEGRAL DOMAIN THE CHARACTERISTIC OF A FIELD A REMINDER OF SOME GROUP THEORY 2 INTEGRAL DOMAINS AND POLYNOMIALS EUCLIDEAN DOMAINS UNIQUE FACTORISATION POLYNOMIALS IRREDUCIBLE POLYNOMIALS 3 FIELD EXTENSIONS THE DEGREE OF AN EXTENSION EXTENSIONS AND POLYNOMIALS POLYNOMIALS AND EXTENSIONS 4 APPLICATIONS TO GEOMETRY RULER AND COMPASSES CONSTRUCTIONS AN ALGEBRAIC APPROACH 5 SPLITTING FIELDS 6 FINITE FIELDS 7 THE GALOIS GROUP MONOMORPHISMS BETWEEN FIELDS AUTOMORPHISMS, GROUPS AND SUB?ELDS NORMAL EXTENSIONS SEPARABLE EXTENSIONS THE GALOIS CORRESPONDENCE THE FUNDAMENTAL THEOREM AN EXAMPLE 8 EQUATIONS AND GROUPS QUADRATICS, CUBICS AND QUARTICS: SOLUTION BY RADICALS CYCLOTOMIC POLYNOMIALS CYCLIC EXTENSIONS 9 SOME GROUP THEORY ABELIAN GROUPS SYLOW SUBGROUPS PERMUTATION GROUPS PROPERTIES OF SOLUBLE GROUPS 10 GROUPS AND EQUATIONS 10 1 INSOLUBLE QUINTICS 10 2 GENERAL POLYNOMIALS 10 3 WHERE NEXT? 11 REGULAR POLYGONS 11 1 PRELIMINARIES 11 2 THE CONSTRUCTION OF REGULAR POLYGONS 12 SOLUTIONS