By Lindsay N. Childs

This publication is written as an advent to better algebra for college students with a heritage of a yr 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 adventure within the algebraic thought of the integers and polynomials to appre ciate the elemental recommendations of summary algebra. the most theoretical thread is to enhance algebraic houses of the hoop of integers: certain 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 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 huge number of functions, to cryptography, error-correcting codes, Latin squares, tournaments, concepts of integration, and particularly to elemen tary and computational quantity idea. A scholar who asks, "Why am I studying this?," willfind solutions frequently inside of a bankruptcy or . For a primary direction in algebra, the ebook bargains a number of merits. • through development the algebra out of numbers and polynomials, the booklet takes maximal benefit of the student's previous adventure in algebra and mathematics. New innovations come up in a well-known context.

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P« and also a = ql .. . qm are factorizations of a into products of primes. We want to show that the two factorizations are the same. If a = PI is prime, then both m and n = 1 and PI = ql, since a prime cannot factor into a product of two or more primes, by definition. So the theorem is true if a is prime. Let us assume that a is not prime. Suppose that PI' the leftmost prime in the first factorization of a, is equal to some prime qj in the second factorization. ) Then a/pi is a natural number ~ 2 (since a is not prime) and, of course, a/pi < a.

E3. Example I, Section A, has a geometric interpreta tion as a way of counting a square array of points. For examp le, for n = 4: I + 3 + 5 + 7 = 42 : o ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 Find and prove the three -dimensional analogue which counts a cubic array of points and for n = 2 gives the formula I + 7 = 8. E4. Pro ve that the sum of the interior angles of an n-sided convex polygon is 1800 x (n - 2). ES. Russian peasant arithmetic. Here is a way of multiplying which has been attributed to Russian peasants who could only add , and multipl y and divide by 2.

E3. In base 2 write the numbers from 29 through 35. Here are some base 60 exercises : E4. One seventh of an hour is 514 seconds, to the nearest second. Write it in minutes and seconds. E5. Write one eleventh of a day in hours, minutes and seconds (to the nearest second). E6. If a runner completes a 50 mile race in 7 hours, 33 minutes and 15 seconds, and he were to run a marathon (26 miles, 385 yards) at the same pace, how long would it take him? (there are 1760 yards to the mile). F. Operations in Base a We can add, subtract, multiply, and divide in any base .