Download e-book for iPad: Algebra. Abstract and Concrete by F. Goodman

By F. Goodman

ISBN-10: 0130673420

ISBN-13: 9780130673428

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We write a b for a C . b/. (d) Multiplication on Z is commutative and associative. (e) 1 is an identity element for multiplication; that is, for all a 2 Z, 1a D a. b C c/ D ab C ac: (g) (h) N is closed under addition and multiplication. That is, the sum and product of positive integers is positive. The product of non-zero integers is non-zero. We write a > b if a b > 0 and a b if a b 0. 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.

As a strategy, let’s compute a few powers of 4 modulo 9. mod 9/. It follows that in Z9 , Œ43k  D Œ43 k D Œ1k D Œ1 for all natural numbers k; likewise, Œ43kC1  D Œ43k Œ4 D Œ4, and Œ43kC2  D Œ43k Œ42 D Œ7. So to compute 4237 modulo 9, we only have to find the conjugacy class of 237 modulo 3; since 237 is divisible by 3, we have Œ4237  D Œ1 in Z9 . mod 7/. The assertion is equivalent to Œa7 D Œa for all Œa 2 Z7 . Now we only have to check this for each of the 7 elements in Z7 , which is straightforward.

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. Thus Œa C b D Œa0 C b 0  and Œab D Œa0 b 0 . This means that it makes sense to define A C B D Œa C b and AB D Œab. 6. Let us look at another example in which we cannot define operations on classes of numbers in the same way (in order to see what the issue is in the preceding discussion).

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Algebra. Abstract and Concrete by F. Goodman

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