1.8. POLYNOMIALS 51 We are about to show that two nonzero polynomials in KŒxŁ always have a greatest common divisor. Notice that a greatest common divisor is unique up to multiplication by a nonzero element of K , by Proposition 1.8.6 (b). There is a unique greatest common divisor that is monic (i.e., whose leading coefﬁcient is 1 ). When we need to refer to the greatest common divisor, we will mean the one that is monic. We denote the monic greatest common divisor of p and q by g.c.d. .p;q/ . The following results ( 1.8.15 through 1.8.21 ) are analogues of results for the integers, and the proofs are virtually identical to those for the in-tegers, with Proposition 1.8.13 playing the role of Proposition 1.6.7 . For each of these results, you should write out a complete proof modeled on the proof of the analogous result for the integers. You will end up under-standing the proofs for the integers better, as well as understanding how they have to be modiﬁed to apply to polynomials. For integers
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