Why Read This Book? - Index of

9.13 Polynomials over the Integers 333
Since f and g are primitive, there are coefficients in both f and g that are
not divisible by p. Let aj and bk be, respectively, the coefficients of the lowest
powers of t in f and g that are not divisible by p; that is, p divides all
of a0,a1,...,aj−1,b0,b1,...,bk−1, but not aj or bk. Then by Exercise 9.13.3,
p does not divide the coefficient of t j+k in fg. Since this is true for all prime
numbers, fg is primitive.
EXERCISE 9.13.3 Finish the proof of Theorem 9.13.2 by showing p does not
divide the coefficient of t j+k in fg. 16
The next theorem says simply that if a polynomial has integer coefficients,
and it can be factored into polynomials of degree at least one by viewing it as
an element of Q[t] and resorting to rational coefficients in the factors, then you
can adjust these factor coefficients and make them integers. We will prove this
in a somewhat conversational way to keep the notation from getting too sloppy.
Notice the point at which we apply Exercise 9.13.3.
Theorem 9.13.4 Suppose f is a polynomial with integer coefficients. If f is
reducible in Q[t], then it is reducible in Z[t].
Proof. Suppose f has integer coefficients, and suppose f = f1f2, where f1 and
f2 are polynomials with rational coefficients, and both f1 and f2 have degree at
least 1. Let d1 and d2 be, respectively, the product of all the denominators of all the
coefficients of f1 and f2; then let g1 = d1f1 and g2 = d2f2, so that g1 and g2 have
integer coefficients. Next, let c1 and c2 be the content of g1 and g2, respectively, and
factor these out to write g1 = c1h1 and g2 = c2h2, so that h1 and h2 are primitive.
Also, if we let c be the content of f , we may write f = cg, where g is primitive.
Thus we have
(cd1d2)g = (d1d2)f = d1f1d2f2 = g1g2 = (c1c2)h1h2
(9.60)
where g, h1, and h2 are all primitive polynomials. By Exercise 9.13.3, h1h2 is
also primitive, so the content of the left-hand side of Eq. (9.60) is cd1d2, and
the content of the right-hand side is c1c2. Since cd1d2 and c1c2 are both positive
integers, cd1d2 = c1c2, and g = h1h2. Thus f = cg = ch1h2, and we have a proper
factorization of f with integer coefficients.
Now the result we have all been waiting for.
Theorem 9.13.5 The factorization in Theorem 9.13.4 is unique up to order and
association of the factors.
16 The coefficient of t j+k is �j+k
i=0 aibj+k−i. Look separately at the terms 0 ≤ i ≤ j − 1, i = j, and
j + 1 ≤ i ≤ j + k.