WORKBOOK IN HIGHER ALGEBRA
CONTENTS GROUP THEORY REVIEW OF IMPORTANT BASICS THE CONCEPT OF A GROUP ACTION SYLOW’S THEOREM EXAMPLES: THE LINEAR GROUPS AUTOMORPHISM GROUPS THE SYMMETRIC AND ALTERNATING GROUPS THE COMMUTATOR SUBGROUP FREE GROUPS; GENERATORS AND RELATIONS FIELD AND GALOIS THEORY BASICS SPLITTING FIELDS AND ALGEBRAIC CLOSURE GALOIS EXTENSIONS AND GALOIS GROUPS SEPARABILITY AND THE GALOIS CRITERION BRIEF INTERLUDE: THE KRULL TOPOLOGY THE FUNDAMENTAL THEOREM OF ALGEBRA THE GALOIS GROUP OF A POLYNOMIAL THE CYCLOTOMIC POLYNOMIALS SOLVABILITY BY RADICALS 2 10 THE PRIMITIVE ELEMENT THEOREM ELEMENTARY FACTORIZATION THEORY UNIQUE FACTORIZATION DOMAINS NOETHERIAN RINGS AND PRINCIPAL IDEAL DOMAINS PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS DEDEKIND DOMAINS A FEW REMARKS ABOUT MODULE THEORY ALGEBRAIC INTEGER DOMAINS OE IS A DEDEKIND DOMAIN FACTORIZATION THEORY IN DEDEKIND DOMAINS THE IDEAL CLASS GROUP OF A DEDEKIND DOMAIN A CHARACTERIZATION OF DEDEKIND DOMAINS MODULE THEORY THE BASIC HOMOMORPHISM THEOREMS DIRECT PRODUCTS AND SUMS OF MODULES MODULES OVER A PRINCIPAL IDEAL DOMAIN CALCULATION OF INVARIANT FACTORS APPLICATION TO A SINGLE LINEAR TRANSFORMATION CHAIN CONDITIONS AND SERIES OF MODULES THE KRULL-SCHMIDT THEOREM INJECTIVE AND PROJECTIVE MODULES SEMISIMPLE MODULES 5 10 EXAMPLE: GROUP ALGEBRAS RING STRUCTURE THEORY THE JACOBSON RADICAL TENSOR PRODUCTS TENSOR PRODUCT AS AN ABELIAN GROUP TENSOR PRODUCT AS A LEFT S-MODULE TENSOR PRODUCT AS AN ALGEBRA TENSOR, SYMMETRIC AND EXTERIOR ALGEBRA THE ADJOINTNESS RELATIONSHIP CHAPTER REVIEW OF IMPORTANT BASICS IN THIS SHORT SECTION WE GATHER TOGETHER SOME OF THE BASICS OF ELEMENTARY GROUP THEORY, AND AT THE SAME TIME ESTABLISH A BIT OF THE NOTATION WHICH WILL BE USED IN THESE NOTES THE FOLLOWING TERMS SHOULD BE WELL-UNDERSTOOD BY THE READER (IF IN DOUBT, CONSULT ANY ELEMENTARY TREATMENT OF GROUP THEORY): 1 GROUP, ABELIAN GROUP, SUBGROUP, COSET, NORMAL SUBGROUP, QUOTIENT GROUP, ORDER OF A GROUP, HOMOMORPHISM, KERNEL OF A HOMOMORPHISM, ISOMORPHISM, NORMALIZER OF A SUBGROUP, CENTRALIZER OF A SUBGROUP, CONJUGACY, INDEX OF A SUBGROUP, SUBGROUP GENERATED BY A SET OF ELEMENTS DENOTE THE IDENTITY ELE- MENT OF THE GROUP G BY E, AND SET G# = G ?{E} IF G IS A GROUP AND IF H IS A SUBGROUP OF G, WE SHALL USUALLY SIMPLY WRITE H ?G HOMOMORPHISMS ARE USUALLY WRITTEN AS LEFT OPERATORS: THUS IF ? : G ?G? IS A HOMOMORPHISM OF GROUPS, AND IF G ?G, WRITE THE IMAGE OF G IN G? AS ?(G) THE FOLLOWING IS BASIC IN THE THEORY OF ?NITE GROUPS THEOREM 1 1 1 (LAGRANGE’S THEOREM) LET G BE A ?NITE GROUP, AND LET H BE A SUBGROUP OF G THEN |H| DIVIDES |G| THE READER SHOULD BE QUITE FAMILIAR WITH BOTH THE STATEMENT, AS WELL AS THE PROOF, OF THE FOLLOWING THEOREM 1 1 2 (THE FUNDAMENTAL HOMOMORPHISM THEOREM) LET G, G? BE GROUPS, AND ASSUME THAT ? : G ?G? IS A SURJECTIVE HOMOMORPHISM 1MANY, IF NOT MOST OF THESE TERMS WILL BE DE?NED BELOW 2 XER R WHERE X1, X2, XR ?X, AND WHERE E1, E2, , ER ?Z IF X = {X}, IT IS CUSTOMARY TO WRITE ?X?IN PLACE OF ?{X}? IF G IS A GROUP SUCH THAT FOR SOME X ?G, G = ?X?, THEN G IS SAID TO BE A CYCLIC GROUP WITH GENERATOR X NOTE THAT, IN GENERAL, A CYCLIC GROUP CAN HAVE MANY GENERATORS THE FOLLOWING CLASSI?ES CYCLIC GROUPS, UP TO ISOMORPHISM: 1 1 REVIEW OF IMPORTANT BASICS 1 LET G BE A GROUP AND LET X ?G BE AN ELEMENT OF ?NITE ORDER N IF K ?Z, SHOW THAT O(XK) = N/(N, K), WHERE (N, K) IS THE GREATEST COMMON DIVISOR OF N AND K CONCLUDE THAT XK IS A GENERATOR OF ?X?IF AND ONLY IF (N, K) = 1 2 LET H, K BE SUBGROUPS OF G, BOTH OF ?NITE INDEX IN G PROVE THAT H ?K ALSO HAS ?NITE INDEX IN FACT, [G : H ?K] = [G : H][H : H ?K]