By Frederick M. Goodman
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Additional info for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)
1. 7 using the well–ordering principle. (a) (b) If a ¤ 0, consider the set S of nonnegative integers that can be written in the form a sd , where s is an integer. Show that S is nonempty. By the well–ordering principle, S has a least element, which we write as r D a qd . Then we have a D qd C r. Show that that r < d. 36 1. 2. 9. m:n/. m; n/, then b divides y. 3. Suppose that a natural number p > 1 has the property that for all nonzero integers a and b, if p divides the product ab, then p divides a or p divides b.
Mod n/, by the previous lemma, part (c). This shows that Œa Â Œb. mod n/, so Œb Â Œa. Thus Œa D Œb. This shows that (a) implies (b). x/ is characterized as the unique element of Œx \ f0; 1; : : : ; n 1g, we have (b) implies (c), and also (c) implies (d). Finally, if Œa \ Œb ¤ ;, let c 2 Œa \ Œb. mod n/. This shows that (d) implies (a). 4. There exist exactly n distinct residue classes modulo n, namely Œ0; Œ1; : : : ; Œn 1. These classes are mutually disjoint. 5. mod n/. mod n/. Proof.
Monthly, 93 (1986), no. 5, 333–348. 22 1. ALGEBRAIC THEMES Continue in this way until has been written as a product of disjoint cycles. 8 In the preceding explanation, the phrase “continue in this way” is a signal that to formalize the argument it is necessary to use induction. 13), uniqueness in the following statement means uniqueness up to order; the factors are unique, and the order in which the factors are written is irrelevant. a2 ; : : : ; ak ; a1 /, and there is no preferred first entry in the cycle notation.
Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) by Frederick M. Goodman