By Joseph Gallian

ISBN-10: 0547165099

ISBN-13: 9780547165097

Modern summary Algebra 7/e presents a superb creation to the conventional subject matters in summary algebra whereas conveying to scholars that it's a modern topic used day-by-day through operating mathematicians, machine scientists, physicists, and chemists. The textual content comprises a number of figures, tables, pictures, charts, biographies, machine routines, and instructed readings giving the topic a present think which makes the content material fascinating and suitable for college students.

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**Example text**

Let a9 be an inverse of a. Then, multiplying on the right by a9 yields (ba)a9 5 (ca)a9. Associativity yields b(aa9) 5 c(aa9). Then, be 5 ce and, therefore, b 5 c as desired. Similarly, one can prove that ab 5 ac implies b 5 c by multiplying by a9 on the left. A consequence of the cancellation property is the fact that in a Cayley table for a group, each group element occurs exactly once in each row and column (see Exercise 23). Another consequence of the cancellation property is the uniqueness of inverses.

9. If a and b are integers and n is a positive integer, prove that a mod n 5 b mod n if and only if n divides a 2 b. 10. Let a and b be integers and d 5 gcd(a, b). If a 5 da9 and b 5 db9, show that gcd(a9, b9) 5 1. 11. Let n be a fixed positive integer greater than 1. If a mod n 5 a9 and b mod n 5 b9, prove that (a 1 b) mod n 5 (a9 1 b9) mod n and (ab) mod n 5 (a9b9) mod n. ) 12. Let a and b be positive integers and let d 5 gcd(a, b) and m 5 lcm(a, b). If t divides both a and b, prove that t divides d.

For every positive integer n, prove that 1 1 2 1 ? ? 1 n 5 n(n 1 1)/2. 23. For every positive integer n, prove that a set with exactly n elements has exactly 2n subsets (counting the empty set and the entire set). 24. For any positive integer n, prove that 2n 32n 2 1 is always divisible by 17. 25. Prove that there is some positive integer n such that n, n 1 1, n 1 2, ? ? , n 1 200 are all composite. 0 | Preliminaries 23 26. (Generalized Euclid’s Lemma) If p is a prime and p divides a1a2 ? ? an, prove that p divides ai for some i.

### Contemporary Abstract Algebra by Joseph Gallian

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