By Frederick M. Goodman
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Extra info for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)
Putting q D q 0 C m, we have p D qd C r. 14. A polynomial f 2 KŒx is a greatest common divisor of nonzero polynomials p; q 2 KŒx if (a) f divides p and q in KŒx and (b) whenever g 2 KŒx divides p and q, then g also divides f . 8. POLYNOMIALS 51 We are about to show that two nonzero polynomials in KŒx always have a greatest common divisor. 6 (b). , whose leading coefficient is 1). When we need to refer to the greatest common divisor, we will mean the one that is monic. p; q/. 7. For each of these results, you should write out a complete proof modeled on the proof of the analogous result for the integers.
Q 0 C 1/d C r, and we are done. We deal with the case a < 0 by induction on jaj. If d < a < 0, take q D 1 and r D aCd . Suppose that a Ä d , and assume inductively that the existence assertion holds for all nonpositive integers whose absolute values are strictly smaller than jaj. a C d / D q 0 d C r and 0 Ä r < d . q 0 1/d C r. So far, we have shown the existence of q and r with the desired properties. For uniqueness, suppose that a D qd C r, and a D q 0 d C r 0 , with 0 Ä r; r 0 < d . q 0 q/d , so r r 0 is divisible by d .
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. Thus Œa C b D Œa0 C b 0 and Œab D Œa0 b 0 .
Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) by Frederick M. Goodman