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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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 Fall '08
 EVERAGE
 Algebra, Polynomials, Multiplication

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