From Polynomials to Sums of Squares

From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.

PrefaceSoftware Copyright and Site LicensePOLYNOMIALS IN ONE VARIABLEPolynomials with rational coefficientsPolynomials with coefficients in Z^O^IpPolynomial divisionCommon divisors of polynomialsUnits, irreducibles and the factor theoremFactorization into irreducible polynomialsPolynomials with integer coefficientsFactorization in Z^O^Ip[^Ix] and applications to Z[^Ix]Factorization in Q[^Ix]Factorizing with the aid of the computerSummary of chapter 1Exercises for chapter 1USING POLYNOMIALS TO MAKE NEW NUMBER FIELDSRoots of irreducible polynomialsThe splitting field of ^Ix^Tpn - ^Ix in Z^O^Ip[^Ix]Summary of chapter 2Exercises for chapter 2QUADRATIC INTEGERS IN GENERAL AND GAUSSIAN INTEGERS IN PARTICULARAlgebraic numbersAlgebraic integersQuadratic numbers and quadratic integersThe integers of Q((square root) -1)Division with remainder in Z[^Ii]Prime and composite integers in Z[^Ii]Summary of chapter 3Exercises for chapter 3ARITHMETIC IN QUADRATIC DOMAINSMultiplicative normsApplication of norms to units in quadratic domainsIrreducible and prime quadratic integersEuclidean domains of quadratic integersFactorization into irreducible integers in quadratic domainsSummary of chapter 4Exercises for chapter 4COMPOSITE RATIONAL INTEGERS AND SUMS OF SQUARESRational primesQuadratic residues and the Legendre symbolIdentifying the rational primes that become composite in a quadratic domainSums of squaresSummary of chapter 5Exercises for chapter 5APPENDICESAbstract perspectivesGroupsRings and integral domainsDivisibility in integral domainsEuclidean domains and factorization into irreduciblesUnique factorization in Euclidean domainsIntegral domains and fieldsFinite fieldsThe product of primitive polynomialsThe M^D"obius function and cyclotomic polynomialsRouch^D'es theoremDirichlet's theorem and Pell's equationQuadratic reciprocityREFERENCESINDEX

T.H. Jackson