Why Read This Book? - Index of

332 Chapter 9 Rings
EXERCISE 9.12.11 Apply Theorem 9.12.10 to the following polynomials
in Z3[t] by either finding a proper factorization or explaining why they are
irreducible.
(a) f1 = t2 + t + 1
(b) f2 = t2 + t + 2
(c) f3 = t3 + t2 + 2
(d) f4 = t3 + t + 2
9.13 Polynomials over the Integers
We have waited to show that Z[t] is a UFD, and now is the time to tackle the
question. We could have shown the existence of a factorization of a polynomial
in Z[t] into irreducibles earlier, but uniqueness of this factorization up to
order and association of the factors requires us to view elements of Z[t] as elements
of Q[t], where factorizations are unique up to association. The reason
we have some work to do to show uniqueness is that reducibility and association
in Q[t] are different from reducibility and association in Z[t]. Polynomials
such as 2t + 6 and 10t + 15 are irreducible and associates in Q[t], but not in Z[t],
because the constant polynomials 2 and 5 are units in Q[t] but not in Z[t]. First the
easy part.
EXERCISE 9.13.1 Every nonzero, non-unit polynomial in Z[t] has a factorization
into irreducible polynomials in Z[t]. 15
To make our way to uniqueness up to order and association, we need the
following lemma. You will provide the climactic detail as an exercise. Remember
that the term primitive applies only to polynomials of degree at least one.
Theorem 9.13.2 (Gauss’s Lemma). In Z[t], the product of two primitive
polynomials is primitive.
Proof. Suppose f and g are primitive polynomials in Z[t]. We show that fg is
primitive by supposing p is any prime number, and then showing there is some
coefficient in fg that is not divisible by p.
Suppose p is a prime number, and write
f = amt m +···+a0 and g = bnt n +···+b0 (9.59)
15 Use strong induction on the degree of f and mimic the proof of Theorem 3.5.19. Exercise 9.10.1 takes
care of the case deg(f ) = 0.