Basic Algebra

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 φ 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. 1. Division and Euclidean Algorithms The first three sections give a careful proof of unique factorization for integers and for polynomials with rational or real or complex coefficients, and they give an indication of some first consequences of this factorization. For the moment let us restrict attention to the set Z of integers. We take addition, subtraction, and multiplication within Z as established, as well as the properties of the usual ordering in Z. A factor of an integer n is a nonzero integer k such that n = kl for some integer l. In this case we say also that k divides n, that k is a divisor of n, and that n is amultiple of k. We write k | n for this relationship. If n is nonzero, any product formula n = kl1 · · · lr is a factorization of n. A unit in Z is a divisor