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3.1. Irreducible Polynomials 81

factor indefinitely, and after a finite number of factorizations must arrive

at factors which are divisible only by constants and constant multiples of

themselves.

Thus, the notion of primeness we require is embodied in this definition:

a polynomial p over an integral domain is irreducible if and only if

(i) degp 2 1 and

(ii) if p = fg for polynomials over the domain, then either f or g is

constant.

Whether or not a polynomial can be factored is sensitive to the domain
over which it is taken. For example, t2 + 1 can be factored as the product
(t-i)(t+i) over the complex field, but it turns out to be irreducible over the
reals and the rationals. Thus, we have to consider each domain individually.
However, the fact that Z C Q C R C C means that factorization with
respect to one of these domains will have some bearing on factorization
with respect to the others.
The study of the solvability of polynomial equations involves looking at
fields in which their roots can be found. Knowing the irreducible factors of
the polynomials enables us to examine the structure of these fields.

Exercises

1. (a) Show that the rational l/2 can be written as the product of two
   rationals in infinitely many ways.
   (b) Show that any element of Q can be written as the product of
   two others in infinitely many ways. (Thus, a field can have no
   prime or irreducible elements.)


2. Let p(t) E Z[t]. Sh ow that the constant polynomial c divides p(t) if
   and only if c divides every coefficient of p(t).


3. Prove that every linear polynomial at + b over an integral domain is
   irreducible.


4. Show that every irreducible polynomial over C is irreducible over R
   and that every irreducible polynomial over R is irreducible over Q.


5. Show that the polynomial t2 + 1 is irreducible over R.


6. Let p(t) be any polynomial over an integral domain and let k belong
   to the domain. Define q(t) = p(t - k). Show that the polynomial q(t)
   is irreducible over the domain if and only if p(t) is irreducible over
   the domain.


7. Let c be an integer.