By Goodman F.M.

ISBN-10: 0979914205

ISBN-13: 9780979914201

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5. mod n/. mod n/. Proof. a C b/ a0 and b b 0 are divisible by n. a is divisible by n. b a0 b 0 / b0/ ■ 40 1. ALGEBRAIC THEMES We denote by Zn the set of residue classes modulo n. The set Zn has a natural algebraic structure which we now describe. Let A and B be elements of Zn , and let a 2 A and b 2 B; we say that a is a representative of the residue class A, and b a representative of the residue class B. The class Œa C b and the class Œab are independent of the choice of representatives. 5.

12. Suppose a is not relatively prime to n. Then there do not exist integers s and t such that as C nt D 1. What does this say about the invertibility of Œa in Zn ? 13. Suppose that Œa is not invertible in Zn . Consider the left multiplication map LŒa W Zn ! Œb/ D ŒaŒb D Œab. Since Œa is not invertible, Œ1 is not in the range of LŒa , so LŒa is not surjective. Conclude that LŒa is not injective, and use this to show that there exists Œb ¤ Œ0 such that ŒaŒb D Œ0 in Zn . 14. Suppose a is relatively prime to n.

Then > is a total order on the integers. That is, it is never true that a > a; for distinct integers a; b, either a > b or b > a; and whenever a > b and b > c, it follows that a > c. 1 is that jabj maxfjaj; jbjg for non-zero integers a and b. 26 1. ALGEBRAIC THEMES Although the multiplicative structure of the integers is subordinate to the additive structure, many of the most interesting properties of the integers have to do with divisibility, factorization, and prime numbers. Of course, these concepts are already familiar to you from school mathematics, so the emphasis in this section will be more on a systematic, logical development of the material, rather than on exploration of unknown territory.

### Algebra. Abstract and concrete by Goodman F.M.

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