Download PDF by Frederick M. Goodman: Algebra: Abstract and Concrete (Stressing Symmetry) (2.5

By Frederick M. Goodman

ISBN-10: 0979914205

ISBN-13: 9780979914201

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Extra resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)

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C) Divisibility is transitive: If ajb and bjc, then ajc. (d) If ajb and ajc, then a divides all integers that can be expressed in the form sb C t c, where s and t are integers. Proof. For part (a), note that neither u nor v can be zero. Suppose first that both are positive. We have uv maxfu; vg 1. If equality holds, then u D v D 1. In general, if uv D 1, then also jujjvj D 1, so juj D jvj D 1. Thus both u and v are ˙1. Since their product is positive, both have the same sign. For (b), let u; v be integers such that b D ua and a D vb.

The first of the permutations takes 1 to 4 and the second takes 4 to 7, so the product takes 1 to 7. The first leaves 7 fixed and the second takes 7 to 6, so the product takes 7 to 6. The first takes 6 to 5 and the second takes 5 to 4, so the product takes 6 to 4. The first takes 4 to 2 and the second leaves 2 fixed, so the product takes 4 to 2. The first takes 2 to 3 and the second takes 3 to 1, so the product takes 2 to 1. 1 7 6 4 2/. The first permutation takes 5 to 6 and the second takes 6 to 5, so the product fixes 5.

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. Thus Œa C b D Œa0 C b 0  and Œab D Œa0 b 0 .

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Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) by Frederick M. Goodman

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