## From Polynomials to Sums of SquaresFrom 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. |

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### Contents

Preface | 7 |

Summary of chapter 1 | 33 |

Summary of chapter 2 | 56 |

Summary of chapter 3 | 80 |

Summary of chapter 4 | 105 |

Summary of chapter 5 | 134 |

The product of primitive polynomials | 155 |

Quadratic reciprocity | 174 |

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### Common terms and phrases

Activity addition and multiplication algebraic integer algebraic numbers Appendix arithmetic composite integers deg(g division with remainder domain of integers Eisenstein's criterion equation Euclid's algorithm Euclidean domain Example expressed extension field finite field finite number form a2 Gaussian integers gives greatest common divisor implies infinitely instance integer coefficients integers in Q(Vd integral domain irreducible elements irreducible factors irreducible in Q[x irreducible integers irreducible polynomial leading coefficient Legendre symbol linear combinations linear factors minimal polynomial modulo x2 natural number non-unit non-zero element odd prime polynomial of degree possible press ENTER prime property primes congruent primitive polynomial product of irreducibles prove Q[jc quadratic domain quadratic integers quadratic residue rational integers rational number rational primes result ring satisfy Similarly smaller degree smaller norm solutions splitting field square modulo Suppose unique factorization unit multiples write written zero Zp[x