84 3. Factors and Zeros
Show that f(t) is irreducible. (Eugen Netto, Mufhematische Annalen
48 (1897)).- Let r be a positive integer. Show that t’ - 2 is irreducible over Q. Is
the polynomial irreducible over R? - For each pair r, n of positive integers with 2 5 r 2 n + 1, show that
there is a polynomial of degree n irreducible over Q with exactly r
terms. - Let p be prime integer. Show that the polynomial tJ’-r + tp-’ +... +
t+ 1 does not satisfy the Eisenstein Criterion for any prime, but that
it is transformed by the substitution t = 1 + s to a polynomial in
s which does satisfy the Criterion for the prime p. Deduce that the
polynomial is irreducible. Check the details when p is given the values
3, 5, and 7. - Find the smallest three integers n for which the polynomial t”-’ +
p-2 +... + t + 1 is reducible. - Suppose f(t) and g(t) are nonzero polynomials over an integral do-
main D contained in C, and that g(t) is irreducible over D. Suppose
also that there is a complex number w (not necessarily in D) for
which f(w) = g(w) = 0. P rove that, for some polynomial h(t) over
D, f(t) = !ww - Let g(t) be an irreducible polynomial over an integral domain D con-
tained in C. Prove that every complex zero of g(t) is simple.
3.2 Strategies for Factoring Polynomials over Z
It is useful to know how to factor polynomials. Not only is factoring helpful
in solving equations, but it is often possible to read off information from a
factorization which would otherwise be hidden.
Even when it is known that a polynomial over Z is reducible, it can be
quite a challenge to actually obtain its factors. While there are systematic
ways of doing so, they are generally complicated and long. Consequently,
it is desirable to have a variety of techniques to make the task manageable.
The exercises in this section will introduce some of these. With experience,
these techniques can be used with discretion and flexibility for efficient
factorization.Exercises
- (a) Consider the quadratic polynomial 6t2 + 2t - 20. Determine two
integers u and v for which u + v = 2 and uv = -120. Verify that