A Concrete Introduction to Higher Algebra by Lindsay N. Childs

By Lindsay N. Childs

This ebook is written as an creation to better algebra for college students with a historical past of a 12 months of calculus. the 1st variation of this ebook emerged from a collection 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 booklet, is to provide scholars sufficient event within the algebraic idea of the integers and polynomials to appre­ ciate the fundamental innovations of summary algebra. the most theoretical thread is to improve algebraic houses of the hoop of integers: particular factorization into primes, congruences and congruence sessions, Fermat's theorem, the chinese language the rest theorem; after which back for the hoop of polynomials. Doing so results in the research of straightforward box extensions, and, particularly, to an exposition of finite fields. trouble-free homes of jewelry, fields, teams, and homomorphisms of those gadgets are brought and used as wanted within the improvement. simultaneously with the theoretical improvement, the booklet provides a large number of purposes, to cryptography, error-correcting codes, Latin squares, tournaments, thoughts of integration, and particularly to elemen­ tary and computational quantity concept. A pupil who asks, "Why am I studying this?," willfind solutions often inside of a bankruptcy or . For a primary direction in algebra, the booklet deals a number of benefits. • through development the algebra out of numbers and polynomials, the publication takes maximal benefit of the student's earlier event in algebra and mathematics. New recommendations come up in a well-recognized context.

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Divide (1, 4,25,46)60 by (1, 38)60' using long division in base 60. Then multiply both numbers by 32 and do the division. Is it any easier? C. but remains one of the fastest algorithms available in modern computational number theory. In this chapter we look at Euclid's algorithm and its immediate consequences, analyze its speed, and, as an interesting aside, discuss a speculative connection between Euclid's algorithm and classical Greek proofs that ~ is irrational when n is not a square. A. Greatest Common Divisors The starting point for the theory in this book is an algorithm of Euclid which describes how to find the greatest common divisor of two natural numbers.

But not for you, with modern notation: show that is irrational. Ji9 E4. " in the paragraph above El. Having done so, explain why the rest of the argument works to show that for j; and 1, the se- 46 3. Euclid's Algorithm quence of ratios So, SI' all large n, SnH = Sn ' E5. In E4, is k always S2' ••• eventually repeats: there is some k > 0 so that for s 2 Lj;J? E6. Try playing the Game of Euclid. Two players play, starting with two natural numbers. The first player subtracts any positive multiple of the lesser of the two numbers from the greater of the two numbers, except that the resulting number must be nonnegative.

By induction, for unique integers rI ' r2 , ••• , rn, 0 ::; ri < a. Then with all ri satisfying 0 ::; r, < a. The expression for b is unique because q and ro are unique in the division theorem. That completes the proof. 0 Notice that the proof shows how to get b in base a. First divide b by a, then, successively, divide the quotients by a: b = aq + ro, 22 2. Induction qn-l = a'O + 'n' The process stops when a quotient is reached which is O. The digits are the 'O)a' remainders: b = ('n'n-l . Here is an example.

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