Chapter I. Preliminaries about the Integers, Polynomials, and Matrices

Abstract

This chapter is mostly a review, discussing unique factorization of positive integers, unique factorization of polynomials whose coefficients are rational or real or complex, signs of permutations, and matrix algebra.

Sections 1–2 concern unique factorization of positive integers. Section 1 proves the division and Euclidean algorithms, used to compute greatest common divisors. Section 2 establishes unique factorization as a consequence and gives several number-theoretic consequences, including the Chinese Remainder Theorem and the evaluation of the Euler $\varphi$ function.

Section 3 develops unique factorization of rational and real and complex polynomials in one indeterminate completely analogously, and it derives the complete factorization of complex polynomials from the Fundamental Theorem of Algebra. The proof of the fundamental theorem is postponed to Chapter IX.

Section 4 discusses permutations of a finite set, establishing the decomposition of each permutation as a disjoint product of cycles. The sign of a permutation is introduced, and it is proved that the sign of a product is the product of the signs.

Sections 5–6 concern matrix algebra. Section 5 reviews row reduction and its role in the solution of simultaneous linear equations. Section 6 defines the arithmetic operations of addition, scalar multiplication, and multiplication of matrices. The process of matrix inversion is related to the method of row reduction, and it is shown that a square matrix with a one-sided inverse automatically has a two-sided inverse that is computable via row reduction.