NUMBER AND ITS ALGEBRA
CONTENTS DIALING IN PROBLEMS DIALING IN SET 1 DIALING IN SET 2 DIALING IN SET 3 DIALING IN SET 4 DIALING IN SET 5 DIALING IN SET 6 DIALING IN SET 7 DIALING IN SET 8 POLYGONS AND MODULAR ARITHMETIC THE COMPLEX NUMBERS THE PENTAGON, GAUSS, AND KRONECKER A THEOREM OF KRONECKER SOME PROPERTIES OF ALGEBRAIC EXTENSIONS OF FIELDS NOW WE CAN SHOW THAT QÐFNÞ IS A FIELD A CRITERION FOR IRREDUCIBILITY MODULAR ARITHMETIC QUADRATIC RECIPROCITY SUPPLEMENT: DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC PROGRESSION A LITTLE GROUP THEORY ORBITS AND ELEMENTARY GROUP THEORY THE FUNDAMENTAL THEOREM OF ARITHMETIC GETTING STARTED COMPUTING GREATEST COMMON DIVISORS MODULAR ARITHMETIC WITH POLYNOMIALS THE GAUSSIAN INTEGERS THE TWO SQUARE THEOREM FERMAT’S LAST THEOREM FORMAL DIRICHLET SERIES AND THE NUMBER OF REPRESENTATIONS OF AN INTEGER AS THE SUM OF TWO SQUARES FORMAL DIRICHLET SERIES SUPPLEMENT: HILBERT’S 17TH PROBLEM THE FUNDAMENTAL THEOREM OF ALGEBRA BACKGROUND FROM ELEMENTARY ANALYSIS FIRST PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA: AN ANALYTIC APPROACH BACKGROUND FROM THE THEORY OF EQUATIONS SECOND PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA: ALL ALGEBRA (ALMOST) THE IDEA BEHIND THE PROOF OF THEOREM 4 11: MORE MODULAR ARITHMETIC WITH POLYNOMIALS GALOIS THEORY AND THE FUNDAMENTAL THEOREM OF ALGEBRA THE TOPOLOGICAL POINT OF VIEW SUPPLEMENT: XN 1 AND ITS FACTORS IRRATIONAL, ALGEBRAIC, AND TRANSCENDENTAL NUMBERS LIOUVILLE’S OBSERVATION GELFOND–SCHNEIDER AND LINDEMANN–WEIERSTRASS THE IRRATIONALITY OF E THE IRRATIONALITY OF P AND EC (C 2 Z) NEXT UP: EC THE TRANSCENDENCE OF E P IS TRANSCENDENTAL MORE ABOUT SYMMETRIC FUNCTIONS EULER’S IDENTITY SETTING THE STAGE AND NOW THE PROOF FOURIER SERIES AND GAUSS SUMS THE FOURIER SERIES OF A DIFFERENTIABLE FUNCTION AND FÐ2Þ DIRICHLET’S THEOREM APPLICATIONS TO NUMERICAL SERIES GAUSS SUMS A BRIEF REVIEW OF IN?NITE INTEGRALS USING COMPLEX NUMBERS THE VALUE OF THE GAUSS SUM ON R SIN KX K EPILOGUE