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Unformatted text preview: . The formal proof goes by induction on the degree of p . If deg .p/ < deg .d/ , then put q D and r D p . So assume now that deg .p/ deg .d/ , and that the result is true when p is replaced by any polynomial of lower degree. According to the lemma, we can write p D md C p , where deg .p / < deg .p/ . By the induction hypothesis, there exist polynomials q and r with deg .r/ < deg .d/ such that p D q d C r . Putting q D q C m , we have p D qd C r . n Denition 1.8.14. A polynomial f 2 Kx is a greatest common divisor of nonzero polynomials p;q 2 Kx if (a) f divides p and q in Kx and (b) whenever g 2 Kx divides p and q , then g also divides f ....
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- Fall '08