Integers, Rational Numbers, and Fields

• Published in: Computer Algebra and Symbolic Computation pp. 35 - 80
• Main Author: Cohen, Joel S.
• Format: Book Chapter
• Language: English
• Published: United Kingdom A K Peters/CRC Press 2003; CRC Press LLC
• Subjects: Multivariate Polynomials; Field Axioms; P U; Chinese Remainder Theorem; Euclidean Algorithm; Mathematical Induction; Additive Inverse; Greatest Common Divisor; Standard Form; Rational Number; Multiplicative Inverse; Function Notation; Extended Euclidean Algorithm; Mathematical Expressions; Computer Algebra Systems; Computer Algebra; Finite Fields; Automatic Simplification; Algebraic Number; Infix Notation; Algebraic Number Fields
• ISBN: 9781568811598; 1568811594
• DOI: 10.1201/9781439863701-8

The chapter is concerned with the numerical objects that arise in computer algebra including the integers, the rational numbers, and other classes of numerical expressions. In Section 2.1 we discuss the basic mathematical properties of the integers and describe some algorithms that are important for computer algebra. Section 2.2 is concerned with the manipulation of rational numbers. We define a standard form for a rational number and describe an algorithm that evaluates involved arithmetic expressions with integers and fractions to a rational number in standard form. In Section 2.3 we introduce the concept of a field, which is a mathematical system with axioms that describe in a general way the algebraic properties of the rational numbers and other classes of expressions that arise in computer algebra. We give a number of examples of fields and show that many transformations that are routinely used in the manipulation of mathematical expressions are logical consequences of the field axioms.