ELEMENTARY ALGEBRA

By C. GIIRE

CONTENTS I TRANSITION TO MODERN NUMBER THEORY 1 HISTORICAL BACKGROUND 2 QUADRATIC RECIPROCITY 3 EQUIVALENCE AND REDUCTION OF QUADRATIC FORMS 4 COMPOSITION OF FORMS, CLASS GROUP 5 GENERA 6 QUADRATIC NUMBER FIELDS AND THEIR UNITS 7 RELATIONSHIP OF QUADRATIC FORMS TO IDEALS 8 PRIMES IN THE PROGRESSIONS 4N + 1 AND 4N + 9 DIRICHLET SERIES AND EULER PRODUCTS 10 DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS 11 PROBLEMS II WEDDERBURN–ARTIN RING THEORY 1 HISTORICAL MOTIVATION 2 SEMISIMPLE RINGS AND WEDDERBURN’S THEOREM 3 RINGS WITH CHAIN CONDITION AND ARTIN’S THEOREM 4 WEDDERBURN–ARTIN RADICAL 5 WEDDERBURN’S MAIN THEOREM 6 SEMISIMPLICITY AND TENSOR PRODUCTS 7 SKOLEM–NOETHER THEOREM 8 DOUBLE CENTRALIZER THEOREM 9 WEDDERBURN’S THEOREM ABOUT FINITE DIVISION RINGS 10 FROBENIUS’S THEOREM ABOUT DIVISION ALGEBRAS OVER THE REALS III BRAUER GROUP 1 DE?NITION AND EXAMPLES, RELATIVE BRAUER GROUP 2 FACTOR SETS 3 CROSSED PRODUCTS 4 HILBERT’S THEOREM 5 DIGRESSION ON COHOMOLOGY OF GROUPS 6 RELATIVE BRAUER GROUP WHEN THE GALOIS GROUP IS CYCLIC 7 PROBLEMS IV HOMOLOGICAL ALGEBRA 1 OVERVIEW 2 COMPLEXES AND ADDITIVE FUNCTORS 3 LONG EXACT SEQUENCES 4 PROJECTIVES AND INJECTIVES 5 DERIVED FUNCTORS 6 LONG EXACT SEQUENCES OF DERIVED FUNCTORS 7 EXT AND TOR 8 ABELIAN CATEGORIES 9 PROBLEMS V THREE THEOREMS IN ALGEBRAIC NUMBER THEORY 1 SETTING 2 DISCRIMINANT 3 DEDEKIND DISCRIMINANT THEOREM 4 CUBIC NUMBER FIELDS AS EXAMPLES 5 DIRICHLET UNIT THEOREM 6 FINITENESS OF THE CLASS NUMBER VI REINTERPRETATION WITH ADELES AND IDELES 1 P-ADIC NUMBERS 2 DISCRETE VALUATIONS 3 ABSOLUTE VALUES 4 COMPLETIONS 5 HENSEL’S LEMMA 6 RAMI?CATION INDICES AND RESIDUE CLASS DEGREES 7 SPECIAL FEATURES OF GALOIS EXTENSIONS 8 DIFFERENT AND DISCRIMINANT 9 GLOBAL AND LOCAL FIELDS 10 ADELES AND IDELES VII INFINITE FIELD EXTENSIONS 1 NULLSTELLENSATZ 2 TRANSCENDENCE DEGREE 3 SEPARABLE AND PURELY INSEPARABLE EXTENSIONS 4 KRULL DIMENSION 5 NONSINGULAR AND SINGULAR POINTS 6 IN?NITE GALOIS GROUPS VIII BACKGROUND FOR ALGEBRAIC GEOMETRY 1 HISTORICAL ORIGINS AND OVERVIEW 2 RESULTANT AND BEZOUT’S THEOREM 3 PROJECTIVE PLANE CURVES 4 INTERSECTION MULTIPLICITY FOR A LINE WITH A CURVE 5 INTERSECTION MULTIPLICITY FOR TWO CURVES 6 GENERAL FORM OF BEZOUT’S THEOREM FOR PLANE CURVES 7 GR¨OBNER BASES 8 CONSTRUCTIVE EXISTENCE 9 UNIQUENESS OF REDUCED GR¨OBNER BASES 10 SIMULTANEOUS SYSTEMS OF POLYNOMIAL EQUATIONS IX THE NUMBER THEORY OF ALGEBRAIC CURVES 2 DIVISORS 3 GENUS 4 RIEMANN–ROCH THEOREM 5 APPLICATIONS OF THE RIEMANN–ROCH THEOREM 6 PROBLEMS X METHODS OF ALGEBRAIC GEOMETRY 1 AF?NE ALGEBRAIC SETS AND AF?NE VARIETIES 2 GEOMETRIC DIMENSION 3 PROJECTIVE ALGEBRAIC SETS AND PROJECTIVE VARIETIES 4 RATIONAL FUNCTIONS AND REGULAR FUNCTIONS 5 MORPHISMS 6 RATIONAL MAPS 7 ZARISKI’S THEOREM ABOUT NONSINGULAR POINTS 8 CLASSI?CATION QUESTIONS ABOUT IRREDUCIBLE CURVES 9 AF?NE ALGEBRAIC SETS FOR MONOMIAL IDEALS 10 HILBERT POLYNOMIAL IN THE AF?NE CASE X METHODS OF ALGEBRAIC GEOMETRY (CONTINUED) 11 HILBERT POLYNOMIAL IN THE PROJECTIVE CASE 12 INTERSECTIONS IN PROJECTIVE SPACE 13 SCHEMES 14 PROBLEMS CHAPTER I TRANSITION TO MODERN NUMBER THEORY ABSTRACT THIS CHAPTER ESTABLISHES GAUSS’S LAW OF QUADRATIC RECIPROCITY, THE THEORY OF BINARY QUADRATIC FORMS, AND DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS SECTION 1 OUTLINES HOW THE THREE TOPICS OF THE CHAPTER OCCURRED IN NATURAL SEQUENCE AND MARKED A TRANSITION AS THE SUBJECT OF NUMBER THEORY DEVELOPED A COHERENCE AND MOVED TOWARD THE KIND OF ALGEBRAIC NUMBER THEORY THAT IS STUDIED TODAY SECTION 2 ESTABLISHES QUADRATIC RECIPROCITY, WHICH IS A REDUCTION FORMULA PROVIDING A RAPID METHOD FOR DECIDING SOLVABILITY OF CONGRUENCES X2 ?M MOD P FOR THE UNKNOWN X WHEN P IS PRIME SECTIONS 3–5 DEVELOP THE THEORY OF BINARY QUADRATIC FORMS AX2 + BXY + CY2, WHERE A, B, C ARE INTEGERS THE BASIC TOOL IS THAT OF PROPER EQUIVALENCE OF TWO SUCH FORMS, WHICH OCCURS WHEN THE TWO FORMS ARE RELATED BY AN INVERTIBLE LINEAR SUBSTITUTION WITH INTEGER COEF?CIENTS AND DETERMINANT 1 THE THEOREMS ESTABLISH THE ?NITENESS OF THE NUMBER OF PROPER EQUIVALENCE CLASSES FOR GIVEN DISCRIMINANT, CONDITIONS FOR THE REPRESENTABILITY OF PRIMES BY FORMS OF A GIVEN DISCRIMINANT, CANONICAL REPRESENTA- TIVES OF THE ?NITELY MANY PROPER EQUIVALENCE CLASSES OF A GIVEN DISCRIMINANT, A GROUP LAW FOR PROPER EQUIVALENCE CLASSES OF FORMS OF THE SAME DISCRIMINANT THAT RESPECTS REPRESENTABILITY OF INTEGERS BY THE CLASSES, AND A THEORY OF GENERA THAT TAKES INTO ACCOUNT INEQUIVALENT FORMS WHOSE VALUES CANNOT BE DISTINGUISHED BY LINEAR CONGRUENCES SECTIONS6–7DIGRESSTOLEAPFORWARDHISTORICALLYANDINTERPRETTHEGROUPLAWFORPROPEREQUIVALENCE CLASSES OF BINARY QUADRATIC FORMS IN TERMS OF AN EQUIVALENCE RELATION ON THE NONZERO IDEALS IN THE RING OF INTEGERS OF AN ASSOCIATED QUADRATIC NUMBER ?ELD SECTIONS 8–10 CONCERN DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS SECTION DISCUSSES EULER’S PRODUCT FORMULA FOR ? N=1 N?S AND SHOWS HOW EULER WAS ABLE TO MODIFY IT TO PROVE THAT THERE ARE IN?NITELY MANY PRIMES 4K + 1 AND IN?NITELY MANY PRIMES 4K + 3 SECTION DEVELOPS DIRICHLET SERIES AS A TOOL TO BE USED IN THE GENERALIZATION, AND SECTION 10 CONTAINS THE PROOF OF DIRICHLET’S THEOREM SECTION 8 USES SOME ELEMENTARY REAL ANALYSIS, AND SECTIONS 9–10 USE BOTH ELEMENTARY REAL ANALYSIS AND ELEMENTARY COMPLEX ANALYSIS 1 HISTORICAL BACKGROUND THE PERIOD 1800 TO 1840 SAW GREAT ADVANCES IN NUMBER THEORY AS THE SUBJECT DEVELOPED A COHERENCE AND MOVED TOWARD THE KIND OF ALGEBRAIC NUMBER THEORY THAT IS STUDIED TODAY THE GROUNDWORK HAD BEEN LAID CHIE?Y BY EUCLID, DIOPHANTUS, FERMAT, EULER, LAGRANGE, AND LEGENDRE SOME OF WHAT THOSE PEOPLE DID WAS REMARKABLY INSIGHTFUL FOR ITS TIME, BUT WHAT COLLECTIVELY HAD COME OUT OF THEIR LABORS WAS MORE A COLLECTION OF MISCELLANEOUS RESULTS THAN AN ORGANIZED THEORY IT WAS GAUSS WHO ?RST GAVE DIRECTION AND DEPTH TO THE SUBJECT, BEGINNING WITH 4 = M FOR UNKNOWN INTEGERS X1, X2, X3, X4; Y2 = X(X ?1)(X +1) FOR UNKNOWN INTEGERS X AND Y; AND XN + YN = ZN FOR UNKNOWN INTEGERS X, Y, Z IN EVERY CASE ONE CAN GET AN IMMEDIATE NECESSARY CONDITION ON A SOLUTION BY WRITING THE EQUATION MODULO SOME INTEGER N THE NECESSARY CONDITION IS THAT THE CORRESPONDING CONGRUENCE MODULO N HAVE A SOLUTION FOR EXAMPLE TAKE THE EQUATION X2 + Y2 = P, WHERE P IS A PRIME, AND LET US ALLOW OURSELVES TO USE THE MORE ELEMENTARY RESULTS OF BASIC ALGEBRA WRITING THE EQUATION MODULO P LEADS TO X2 + Y2 ?0 MOD P CERTAINLY X CANNOT BE DIVISIBLE BY P, SINCE OTHERWISE Y WOULD BE DIVISIBLE BY P, X2 AND Y2 WOULD BE DIVISIBLE BY P2, AND X2 + Y2 = P WOULD BE DIVISIBLE BY P2, CONTRADICTION THUS WE CAN DIVIDE, OBTAINING 1 + (YX?1)2 ?0 MOD P HENCE Z2 ??1 MOD P FOR Z ?XY?1 IF P IS AN ODD PRIME, THEN ?1 HAS ORDER 2, AND THE NECESSARY CONDITION IS THAT THERE EXIST SOME Z IN F× P WHOSE ORDER IS EXACTLY 4 SINCE F× P IS CYCLIC OF ORDER P ?1, THE NECESSARY CONDITION IS THAT 4 DIVIDE P ?1 USING A SLIGHTLY MORE COMPLICATED ARGUMENT, WE CAN ESTABLISH CONVERSELY THAT THE DIVISIBILITY OF P ?1 BY 4 IMPLIES THAT X2 + Y2 = P IS SOLVABLE FOR INTEGERS X AND Y IN FACT, WE KNOW FROM THE SOLVABILITY OF Z2 ??1 MOD P THAT THERE EXISTS AN INTEGER R SUCH THAT P DIVIDES R2 + 1 CONSIDER THE POSSIBILITIES IN THE INTEGRAL DOMAIN Z[I] OF GAUSSIAN INTEGERS, WHERE I = ??1 IT WAS SHOWN IN CHAPTER VIII OF BASIC ALGEBRA THAT Z[I] IS EUCLIDEAN HENCE Z[I] IS A PRINCIPAL IDEAL DOMAIN, AND ITS ELEMENTS HAVE UNIQUE FACTORIZATION IF P REMAINS PRIME IN Z[I], THEN THE FACT THAT P DIVIDES (R + I)(R ?I) IMPLIES THAT P DIVIDES R + I OR R ?I IN Z[I] THEN AT LEAST ONE OF R P + I P AND R P ?I P WOULD HAVE TO BE IN Z[I] SINCE I 2) = (X1X2 ?Y1Y2)2 + (X1Y2 + X2Y1)2, 1 HISTORICAL BACKGROUND WHICH HAS BEEN KNOWN SINCE ANTIQUITY, AND WE SEE THAT X2 + Y2 = M IS SOLVABLE IF M IS A PRODUCT OF ODD PRIMES OF THE FORM 4K + 1 IT IS SOLVABLE ALSO IF M = DIVIDING M OCCURS AN EVEN NUMBER OF TIMES IN THE PRIME FACTORIZATION OF M THE ?RST STEP IN THE ABOVE ARGUMENT USED CONGRUENCE INFORMATION; WE HAD TO KNOW THE PRIMES P FOR WHICH Z2 ??1 MOD P IS SOLVABLE THE SECOND STEP WAS IN TWO PARTS—BOTH RATHER SPECIAL FIRST WE USED SPECI?C INFORMATION ABOUT THE NATURE OF FACTORIZATION IN A PARTICULAR RING OF ALGEBRAIC INTEGERS, NAMELY Z[I] SECOND WE USED THAT THE NORM OF A PRODUCT IS THE PRODUCT OF THE NORMS IN THAT SAME RING OF ALGEBRAIC INTEGERS IT IS TOO MUCH TO HOPE THAT SOME RECOGNIZABLE GENERALIZATION OF THESE STEPS WITH X2 + Y2 = M CAN HANDLE ALL OR MOST DIOPHANTINE EQUATIONS AT LEAST THE ?RST STEP IS AVAILABLE IN COMPLETE GENERALITY, AND INDEED NUMBER THEORY—BOTH CLASSICAL AND MODERN—DEDUCES MANY HELPFUL CONCLUSIONS BY PASSING TO CONGRUENCES THERE IS THE MATTER OF DEDUCING SOMETHING USEFUL FROM A GIVEN CONGRUENCE, BUT DOING SO IS A ?NITE PROBLEM FOR EACH PRIME LIKE SOME OTHERS BEFORE HIM, GAUSS SET ABOUT STUDYING CONGRUENCES SYSTEMATICALLY LINEAR CONGRUENCES ARE EASY AND HAD BEEN HANDLED BEFORE QUADRATIC CONGRUENCES ARE LOGICALLY THE NEXT STEP THE ?RST JEWEL OF CLASSICAL NUMBER THEORY TO BE DISCUSSED IN THIS CHAPTER IS THE LAW OF QUADRATIC RECIPROCITY OF GAUSS, WHICH APPEARS BELOW AS THEOREM 1 2 AND WHICH MAKES USEFUL DEDUCTIONS POSSIBLE IN THE CASE OF QUADRATIC CONGRUENCES IN EFFECT QUADRATIC RECIPROCITY ALLOWS ONE TO DECIDE EASILY WHICH INTEGERS ARE SQUARES MODULO A PRIME P EULER HAD EARLIER COME CLOSE TO ?NDING THE STATEMENT OF THIS RESULT, AND LEGENDRE HAD FOUND THE EXACT STATEMENT WITHOUT ?NDING A COMPLETE PROOF GAUSS WAS THE ONE WHO GAVE THE ?RST COMPLETE PROOF PART OF THE UTILITY OF QUADRATIC RECIPROCITY IS THAT IT HELPS ONE TO ATTACK QUADRATIC DIOPHANTINEEQUATIONSMORESYSTEMATICALLY THESECONDJEWELOFCLASSICALNUMBER THEORY TO BE DISCUSSED IN THIS CHAPTER IS THE BODY OF RESULTS CONCERNING REPRESENTING INTEGERS BY BINARY QUADRATIC FORMS AX2+BXY+CY2 = M THAT DO NOT DEGENERATE IN SOME WAY LAGRANGE AND LEGENDRE HAD ALREADY MADE ADVANCES IN THIS THEORY, BUT GAUSS’S OWN DISCOVERIES WERE DECISIVE DIRICHLET SIMPLI?ED THE MORE ADVANCED PARTS OF THE THEORY AND INVESTIGATED AN ASPECT OF IT THAT GAUSS HAD NOT ADDRESSED 2) = (2X1X2 + X1Y2 + Y1X2 + 3Y1Y2)2 + 5(X1X2 ?Y1Y2)2, FROM WHICH IT FOLLOWS THAT THE PRODUCT OF TWO PRIMES CONGRUENT TO 3 OR 7 MODULO 20 IS REPRESENTABLE AS X2 + 5Y2; THIS FACT HAD BEEN CONJECTURED BY FERMAT LEGENDRE ADDED TO THIS INVESTIGATION THE CORRECT FORMULA FOR QUADRATIC RECI- PROCITY, WHICH HE INCORRECTLY BELIEVED HE HAD PROVED, AND MANY OF ITS CONSE- QUENCES FOR REPRESENTABILITY OF PRIMES BY BINARY QUADRATIC FORMS IN ADDITION, HE TRIED TO DEVELOP A THEORY OF COMPOSITION OF FORMS THAT GENERALIZES LAGRANGE’S IDENTITY ABOVE, BUT HE HAD ONLY LIMITED SUCCESS IN ADDITION TO ESTABLISHING QUADRATIC RECIPROCITY, GAUSS INTRODUCED THE VITAL NO- TIONOF “PROPEREQUIVALENCE” FORFORMSAX2+BXY+CY2 OFTHESAMEDISCRIMINANT— TWO FORMS OF THE SAME DISCRIMINANT BEING PROPERLY EQUIVALENT IF ONE CAN BE OBTAINED FROM THE OTHER BY A LINEAR INVERTIBLE SUBSTITUTION WITH INTEGER ENTRIES AND DETERMINANT +1 IN TERMS OF THIS DE?NITION, HE SETTLED THE REPRESENTABILITY OF PRIMES BY BINARY QUADRATIC FORMS, HE SHOWED THAT THERE ARE ONLY ?NITELY MANY PROPER EQUIVALENCES CLASSES FOR EACH DISCRIMINANT, AND HE GAVE AN ALGORITHM FOR 1THESE MATTERS ARE AF?RMED IN DIRICHLET’S LECTURES ON NUMBER THEORY THE ASPECT THAT GAUSS HAD NOT ADDRESSED AND THAT PROVIDED MOTIVATION FOR DIRICHLET IS THE VALUE OF THE “DIRICHLET CLASS NUMBER” H(D) DE?NED BELOW 1 HISTORICAL BACKGROUND WITH THE CONTEXT REPHRASED WE COME BACK TO THE EXAMPLE CONSIDER THE EQUATION X2+5Y2 = P FOR PRIMES P THE DISCRIMINANT OF X2+5Y2 IS ?20, AND THE RELEVANT RING OF ALGEBRAIC INTEGERS IS Z[??5 ], WHICH IS NOT A UNIQUE FACTORIZATION DOMAIN THUS THE ARGUMENT USED WITH X2 + Y2 = P DOES NOT APPLY, AND WE HAVE NO REASON TO EXPECT THAT SOLVABILITY OF X2 + 5Y2 ?0 MOD P IS SUF?CIENT FOR SOLVABILITY OF X2 +5Y2 = P LET US LOOK MORE CLOSELY THE CONGRUENCE CONDITION IS THAT ?20 IS A SQUARE MODULO P THUS ?5 IS TO BE A SQUARE MODULO P IF WE LEAVE ASIDE THE PRIMES P = 2 AND P = 5 THAT DIVIDE 20, THE LAW OF QUADRATIC RECIPROCITY WILL TELL US THAT THE NECESSARY CONGRUENCE RESULTING FROM SOLVABILITY OF X2 + 5Y2 = P IS THAT P BE CONGRUENT TO 1, 3, 7, OR 9 MODULO 20 HOWEVER, WE CAN COMPUTE ALL RESIDUES N OF X2 + 5Y2 MODULO 20 FOR N WITH GCD(N, 20) = 1 TO SEE THAT X2 + 5Y2 ?1 OR 9 MOD IF GCD(X2 + 5Y2, 20) = 1 MEANWHILE, THE FORM 2X2 + 2XY + 3Y2 HAS DISCRIMINANT ?20, AND WE CAN CHECK THAT SOLVABILITY OF 2X2 + 2XY + 3Y2 = P LEADS TO THE CONCLUSION THAT 2X2 + 2XY + 3Y2 ?3 OR 7 MOD