FIELD THEORY
CONTENTS PRELIMINARIES 0 1 LATTICES 0 2 GROUPS 0 3 THE SYMMETRIC GROUP 0 4 RINGS 0 5 INTEGRAL DOMAINS 0 6 UNIQUE FACTORIZATION DOMAINS 0 7 PRINCIPAL IDEAL DOMAINS 0 8 EUCLIDEAN DOMAINS 0 9 TENSOR PRODUCTS EXERCISES PART I—FIELD EXTENSIONS POLYNOMIALS 1 1 POLYNOMIALS OVER A RING 1 2 PRIMITIVE POLYNOMIALS AND IRREDUCIBILITY 1 3 THE DIVISION ALGORITHM AND ITS CONSEQUENCES 1 4 SPLITTING FIELDS 1 5 THE MINIMAL POLYNOMIAL 1 6 MULTIPLE ROOTS 1 7 TESTING FOR IRREDUCIBILITY FIELD EXTENSIONS 2 1 THE LATTICE OF SUBFIELDS OF A FIELD 2 2 TYPES OF FIELD EXTENSIONS 2 3 FINITELY GENERATED EXTENSIONS 2 4 SIMPLE EXTENSIONS 2 5 FINITE EXTENSIONS 2 6 ALGEBRAIC EXTENSIONS 2 7 ALGEBRAIC CLOSURES 2 8 EMBEDDINGS AND THEIR EXTENSIONS 2 9 SPLITTING FIELDS AND NORMAL EXTENSIONS EMBEDDINGS AND SEPARABILITY 3 1 RECAP AND A USEFUL LEMMA 3 2 THE NUMBER OF EXTENSIONS: SEPARABLE DEGREE 3 3 SEPARABLE EXTENSIONS 3 4 PERFECT FIELDS 3 5 PURE INSEPARABILITY *3 6 SEPARABLE AND PURELY INSEPARABLE CLOSURES ALGEBRAIC INDEPENDENCE 4 1 DEPENDENCE RELATIONS 4 2 ALGEBRAIC DEPENDENCE 4 3 TRANSCENDENCE BASES *4 4 SIMPLE TRANSCENDENTAL EXTENSIONS PART II—GALOIS THEORY GALOIS THEORY I: AN HISTORICAL PERSPECTIVE 5 1 THE QUADRATIC EQUATION 5 2 THE CUBIC AND QUARTIC EQUATIONS 5 3 HIGHER-DEGREE EQUATIONS 5 4 NEWTON'S CONTRIBUTION: SYMMETRIC POLYNOMIALS 5 5 VANDERMONDE 5 6 LAGRANGE 5 7 GAUSS 5 8 BACK TO LAGRANGE 5 9 GALOIS 5 10 A VERY BRIEF LOOK AT THE LIFE OF GALOIS GALOIS THEORY II: THE THEORY 6 1 GALOIS CONNECTIONS 6 2 THE GALOIS CORRESPONDENCE 6 3 WHO'S CLOSED? 6 4 NORMAL SUBGROUPS AND NORMAL EXTENSIONS 6 5 MORE ON GALOIS GROUPS 6 6 ABELIAN AND CYCLIC EXTENSIONS *6 7 LINEAR DISJOINTNESS GALOIS THEORY III: THE GALOIS GROUP OF A POLYNOMIAL 7 1 THE GALOIS GROUP OF A POLYNOMIAL 7 2 SYMMETRIC POLYNOMIALS 7 3 THE FUNDAMENTAL THEOREM OF ALGEBRA 7 4 THE DISCRIMINANT OF A POLYNOMIAL 7 5 THE GALOIS GROUPS OF SOME SMALL-DEGREE POLYNOMIALS A FIELD EXTENSION AS A VECTOR SPACE 8 1 THE NORM AND THE TRACE *8 2 CHARACTERIZING BASES *8 3 THE NORMAL BASIS THEOREM FINITE FIELDS I: BASIC PROPERTIES 9 1 FINITE FIELDS REDUX 9 2 FINITE FIELDS AS SPLITTING FIELDS 9 3 THE SUBFIELDS OF A FINITE FIELD 9 4 THE MULTIPLICATIVE STRUCTURE OF A FINITE FIELD 9 5 THE GALOIS GROUP OF A FINITE FIELD 9 6 IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS *9 7 NORMAL BASES *9 8 THE ALGEBRAIC CLOSURE OF A FINITE FIELD FINITE FIELDS II: ADDITIONAL PROPERTIES 10 1 FINITE FIELD ARITHMETIC *10 2 THE NUMBER OF IRREDUCIBLE POLYNOMIALS *10 3 POLYNOMIAL FUNCTIONS *10 4 LINEARIZED POLYNOMIALS THE ROOTS OF UNITY 11 1 ROOTS OF UNITY 11 2 CYCLOTOMIC EXTENSIONS *11 3 NORMAL BASES AND ROOTS OF UNITY *11 4 WEDDERBURN'S THEOREM *11 5 REALIZING GROUPS AS GALOIS GROUPS CYCLIC EXTENSIONS 12 1 CYCLIC EXTENSIONS 12 2 EXTENSIONS OF DEGREE CHAR ²-³ EXERCISES SOLVABLE EXTENSIONS 13 1 SOLVABLE GROUPS 13 2 SOLVABLE EXTENSIONS 13 3 RADICAL EXTENSIONS 13 4 SOLVABILITY BY RADICALS 13 5 SOLVABLE EQUIVALENT TO SOLVABLE BY RADICALS 13 6 NATURAL AND ACCESSORY IRRATIONALITIES 13 7 POLYNOMIAL EQUATIONS PART III—THE THEORY OF BINOMIALS BINOMIALS 14 1 IRREDUCIBILITY 14 2 THE GALOIS GROUP OF A BINOMIAL *14 3 THE INDEPENDENCE OF IRRATIONAL NUMBERS FAMILIES OF BINOMIALS 15 1 THE SPLITTING FIELD 15 2 DUAL GROUPS AND PAIRINGS 15 3 KUMMER THEORY CHAPTER PRELIMINARIES THE PURPOSE OF THIS CHAPTER IS TO REVIEW SOME BASIC FACTS THAT WILL BE NEEDED IN THE BOOK THE DISCUSSION IS NOT INTENDED TO BE COMPLETE, NOR ARE ALL PROOFS SUPPLIED WE SUGGEST THAT THE READER QUICKLY SKIM THIS CHAPTER (OR SKIP IT ALTOGETHER) AND USE IT AS A REFERENCE IF NEEDED 0 1 LATTICES DEFINITION A OR IS A NONEMPTY SET , TOGETHER PARTIALLY ORDERED SET POSET ( ) WITH A BINARY RELATION ON SATISFYING THE FOLLOWING PROPERTIES FOR ALL , , 7, 1) ( ) REFLEXIVITY 2) ( ) ANTISYMMETRY ~ , ¬ 3) ( ) TRANSITIVITY , ¬ IF, IN ADDITION, , OR 7 ¬ THEN IS SAID TO BE TOTALLY ORDERED ANY SUBSET OF A POSET IS ALSO A POSET UNDER THE RESTRICTION OF THE RELATION DEFINED ON A TOTALLY ORDERED SUBSET OF A POSET IS CALLED A IF AND ‹ CHAIN FOR ALL THEN IS CALLED AN FOR A UPPER BOUND LEAST UPPER BOUND FOR , DENOTED BY OR , IS AN UPPER BOUND THAT IS LESS THAN OR : ²:³ : LUB EQUAL TO ANY OTHER UPPER BOUND SIMILAR STATEMENTS HOLD FOR LOWER BOUNDS AND GREATEST LOWER BOUNDS, THE LATTER DENOTED BY , OR A GLB²:³ : W MAXIMAL ELEMENT IN A POSET IS AN ELEMENT SUCH THAT IMPLIES A ~ MINIMAL ELEMENT IN A POSET IS AN ELEMENT SUCH THAT IMPLIES FIELD THEORY ~ A TOP ELEMENT IS AN ELEMENT WITH THE PROPERTY THAT FOR ALL SIMILARLY, A IS AN ELEMENT WITH THE PROPERTY THAT BOTTOM ELEMENT FOR ALL ZORN'S LEMMA SAYS THAT IF EVERY CHAIN IN A POSET HAS AN UPPER BOUND IN THEN HAS A MAXIMAL ELEMENT DEFINITION A IS A POSET IN WHICH EVERY PAIR OF ELEMENTS , HAS A LATTICE LEAST UPPER BOUND, OR , DENOTED BY AND A GREATEST LOWER BOUND, OR JOIN V MEET, DENOTED BY IF EVERY NONEMPTY SUBSET OF HAS A JOIN AND A MEET W THEN IS CALLED A COMPLETE LATTICE NOTE THAT ANY NONEMPTY COMPLETE LATTICE HAS A GREATEST ELEMENT, DENOTED BY AND A SMALLEST ELEMENT, DENOTED BY DEFINITION A OF A LATTICE IS A SUBSET OF THAT IS CLOSED UNDER THE SUBLATTICE MEET AND JOIN OPERATION OF IT IS IMPORTANT TO NOTE THAT A SUBSET OF A LATTICE CAN BE A LATTICE UNDER THE SAME ORDER RELATION AND YET NOT BE A SUBLATTICE OF AS AN EXAMPLE, CONSIDER THE COLL OF ALL SUBGROUPS OF A GROUP , ORDERED BY INCLUSION THEN IS A SUBSET OF THE I I POWER SET , WHICH IS A LATTICE UNDER UNION AND INTERSECTION BUT IS NOT A F I ² ³ SUBLATTICE OF SINCE THE UNION OF TWO SUBGROUPS NEED NOT BE A SUBGROUP F² ³ ON THE OTHER HAND, IS A LATTICE IN ITS OWN RIGHT UNDER SET INCLUSION, WHERE THE I MEET OF TWO SUBGROUPS IS THEIR INTERSECTION AND THE JOIN IS THE / W V SMALLEST SUBGROUP OF CONTAINING AND / IN A COMPLETE LATTICE , JOINS CAN BE DEFINED IN TERMS OF MEETS, SINCE IS THE ; MEET OF ALL UPPER BOUNDS OF THE FACT THAT ENSURES THAT HAS AT LEAST ; ; ONE UPPER BOUND, SO THAT THE MEET IS NOT AN EMPTY ONE THE FOLLOWING THEOREM EXPLOITS THIS IDEA TO GIVE CONDITIONS UNDER WHICH A SUBSET OF A COMPLETE LATTICE IS ITSELF A COMPLETE LATTICE THEOREM 0 1 1 LET BE A COMPLETE LATTICE IF HAS THE PROPERTIES 1) , ) ( ) CLOSED UNDER ARBITRARY MEETS ; ‹ : ; £ J ¬ ; : THEN IS A COMPLETE LATTICE UNDER THE SAME MEET : PROOF LET THEN BY ASSUMPTION LET BE THE SET OF ALL UPPER ; ‹ : ; : < BOUNDS OF THAT LIE IN SINCE , WE HAVE HENCE, AND IS ; : : < £ J < : ; THUS, IS A COMPLETE LATTICE (NOTE THAT NEED NOT BE A SUBLATTICE OF SINCE NEED NOT EQUAL THE MEET OF ALL UPPER BOUNDS OF IN ) < ; 0 2 GROUPS DEFINITION A IS A NONEMPTY SET , TOGETHER WITH A BINARY OPERATION ON GROUP D ¦ , THAT IS, A MAP , DENOTED BY JUXTAPOSITION, WITH THE FOLLOWING PROPERTIES: PRELIMINARIES FOR ALL , , ) ( ) ASSOCIATIVITY ² ³ ~ ² ³ THERE EXISTS AN ELEMENT FOR WHICH FOR ALL ) ( ) IDENTITY ~ ~ FOR EACH , THERE IS AN ELEMENT FOR WHICH ) ( ) INVERSES C C C ~ ~ A GROUP IS , OR , IF , FOR ALL , ~ ABELIAN COMMUTATIVE THE IDENTITY ELEMENT IS OFTEN DENOTED BY WHEN IS ABELIAN, THE GROUP OPERATION IS OFTEN DENOTED BY AND THE IDENTITY BY B SUBGROUPS DEFINITION A OF A GROUP IS A SUBSET OF THAT IS A GROUP IN ITS SUBGROUP : OWN RIGHT, USING THE RESTRICTION OF THE OPERATION DEFINED ON WE DENOTE THE FACT THAT IS A SUBGROUP OF BY WRITING : : IF IS A GROUP AND , THEN THE SET OF ALL POWERS OF º ~ “ { ¸ ¹ IS A SUBGROUP OF