By Lindsay N. Childs

ISBN-10: 0387989994

ISBN-13: 9780387989990

ISBN-10: 1441987029

ISBN-13: 9781441987020

This ebook is written as an creation to better algebra for college students with a history of a 12 months of calculus. the 1st version of this ebook emerged from a suite of notes written within the 1970sfor a sophomore-junior point direction on the college at Albany entitled "Classical Algebra." the target of the direction, and the e-book, is to offer scholars sufficient event within the algebraic conception of the integers and polynomials to appre ciate the elemental strategies of summary algebra. the most theoretical thread is to advance algebraic houses of the hoop of integers: designated factorization into primes, congruences and congruence periods, Fermat's theorem, the chinese language the rest theorem; after which back for the hoop of polynomials. Doing so results in the examine of easy box extensions, and, particularly, to an exposition of finite fields. effortless houses of jewelry, fields, teams, and homomorphisms of those items are brought and used as wanted within the improvement. at the same time with the theoretical improvement, the booklet provides a extensive number of functions, to cryptography, error-correcting codes, Latin squares, tournaments, recommendations of integration, and particularly to elemen tary and computational quantity thought. A pupil who asks, "Why am I studying this?," willfind solutions frequently inside of a bankruptcy or . For a primary direction in algebra, the booklet bargains a number of merits. • through construction the algebra out of numbers and polynomials, the e-book takes maximal benefit of the student's earlier adventure in algebra and mathematics. New techniques come up in a well-recognized context.

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**Example text**

37 D. The Efficiency of Euclid's Algorithm How much more efficient is Euclid's algorithm! It finds that 359 is the last nonzero remainder, and hence is the greatest common divisor of 92263 and 159037, in a total of 10 divisions. In this section we explore how efficient Euclid's algorithm is on any two numbers a, b. Let N(a, b) denote the number of steps needed to obtain the last nonzero remainder of a and b (a ~ b) in Euclid's algorithm using division and not just subtraction. Thus, as the algorithm is laid out in section B, N(a ,b) = n.

Ii) Prove that an+s /an ~ 10 for all n ~ 5. (iii) Prove that a S d + 2 has at least d + 1 decimal digits. Now if a S d + Z has at least d + I decimal digits, then any number a with d digits satisfies a :s;; a S d + Z • So from Lame's theorem we get: Corollary 1. If a < b and a has d digits, then N(a, b) :s;; (5d + 2) - 3 < 5d. The corollary shows how efficient Euclid's algorithm is. Even on the worst possible examples, Euclid's algorithm takes less than 5d steps, where d is the number of decimal digits of the smaller of the two numbers being computed.

E14. You are given two "hour" glasses: a 6-minute hourglass and an ll-minute hourglass, and you wish to measure 13 minutes. How do you do it? E15. You take a 12 quart jug and a 17 quart jug to a stream and want to bring back 8 quarts of water. How do you do it? E16. You take an a quart jug and a b quart jug to the stream and want to bring back e quarts of water. For which e can it be done? How? E17. Show that (a, (b, e)) = ((a, b), c) for any three numbers a, b, e. Define the greatest common divisor of a, b, e, call it (a, b, c), and show that (a, b, c) = (a, (b, e)).

### A Concrete Introduction to Higher Algebra by Lindsay N. Childs

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