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From Polynomials to Sums of Squares [Kõva köide]

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(University of York, UK)
  • Formaat: Hardback, 194 pages, kõrgus x laius: 210x148 mm, kaal: 520 g
  • Ilmumisaeg: 30-Sep-2020
  • Kirjastus: CRC Press
  • ISBN-10: 113845432X
  • ISBN-13: 9781138454323
Teised raamatud teemal:
From Polynomials to Sums of SquaresSuurem pilt
  • Formaat: Hardback, 194 pages, kõrgus x laius: 210x148 mm, kaal: 520 g
  • Ilmumisaeg: 30-Sep-2020
  • Kirjastus: CRC Press
  • ISBN-10: 113845432X
  • ISBN-13: 9781138454323
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781138454323.html
Märksõnad:
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.
Preface -- 1 Polynomials in one variable -- 1.1 Polynomials with
rational coefficients -- 1.2 Polynomials with coefficients in Zp -- 1.3
Polynomial division -- 1.4 Common divisors of polynomials -- 1.5 Units,
irreducibles and the factor theorem -- 1.6 Factorization into irreducible
polynomials -- 1.7 Polynomials with integer coefficients -- 1.8 Factorization
in Zp [ x] and applications to Z[ x] -- 1.9 Factorization in Q[ x] -- 1.10
Factorizing with the aid of the computer -- Summary of chapter 1 -- Exercises
for chapter 1 -- 2 Using polynomials to make new number fields -- 2.1 Roots
of irreducible polynomials -- 2.2 The splitting field of xP" - x in Zp [ x] --
Summary of chapter 2 -- Exercises for chapter 2 -- 3 Quadratic integers in
general and Gaussian integers in particular -- 3.1 Algebraic numbers -- 3.2
Algebraic integers -- 3.3 Quadratic numbers and quadratic integers -- 3.4 The
integers of Q(-J=T) -- 3.5 Division with remainder in Z[ i] -- 3.6 Prime and
composite integers in Z[ i] -- Summary of chapter 3 -- Exercises for chapter 3
-- 4 Arithmetic in quadratic domains -- 4.1 Multiplicative norms -- 4.2
Application of norms to units in quadratic domains -- 4.3 Irreducible and
prime quadratic integers -- 4.4 Euclidean domains of quadratic integers --
4.5 Factorization into irreducible integers in quadratic -- domains --
Summary of chapter 4 -- Exercises for chapter 4 -- 5 Composite rational
integers and sums of squares -- 5.1 Rational primes -- 5.2 Quadratic residues
and the Legendre symbol -- 5.3 Identifying the rational primes that become
composite in a quadratic domain -- 5.4 Sums of squares -- Summary of chapter
5 -- Exercises for chapter 5 -- Appendices -- 1 Abstract perspectives -- 1.1
Groups -- 1.2 Rings and integral domains -- 1.3 Divisibility in integral
domains -- 1.4 Euclidean domains and factorization into irreducibles -- 1.5
Unique factorization in Euclidean domains -- 1.6 Integral domains and fields
-- 1.7 Finite fields -- 2 The product of primitive polynomials -- 3 The
Mobius function and cyclotomic polynomials -- 4 Rouches theorem -- 5
Dirichlet's theorem and Pell's equation -- 6 Quadratic reciprocity --
References Index.
Jackson, T.H