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(^82) 3. Factors and Zeros
(a) Show that, if c is positive, t2 +c is irreducible over Z, Q and R,
but reducible over C.
(b) Suppose that c = -d2 for some integer d. Show that t2 + c is
reducible over 2, Q, R and C.
(c) Suppose that c is negative, but, not the negative of an integer
square. Show that t2+c is irreducible over Z and Q, but reducible
over R and C.

8. Discuss the irreducibility of the polynomial at2 + bt + c over Z, Q, R
   and C, where a, b, c are integers.


9. Show that the polynomial St2 - gt - g is reducible over Q and write
   it as a product of linear factors.


10. Since Q is the smallest field which contains the ring Z, one might
    wonder whether there is some connection between reducibility over Q
    and over Z. Consider first a polynomial whose coefficients are integers;
    obviously, if it is reducible over Z, it is reducible over Q. On the face
    of it, it is not clear that the converse is true. The following exercises
    will deal with this issue.

(a) Let f(t) be a polynomial with rational coefficients. Show that

f(t) can be written in the form (c/d)g(t) where g(t) is a poly-

nomial with integer coefficients whose greatest common divisor

is 1.

(b) If g(t) is reducible over Z, show that f(t) is reducible over Q.

(c) Suppose that f(t) is reducible over Q. Show that g(t) can be

written in the form

g(t) = (albh Wgdt)

where a/b is a fraction in its lowest terms and g1 and g2 are poly-

nomials over Z such that the coefficients of each have greatest

common divisor 1.

(d) If it can be shown, in (c), that lb1 = 1, it will follow that g(t)

is reducible over Z. To obtain a proof by contradiction of this

fact, suppose that p is any prime which divides b. Let uttr and

v,t’ be the terms of lowest degree of g1 and g2 respectively

whose coefficients are not divisible by p. Show that the coefficient

of tr+’ in gl(t)gz(t) is not divisible by p, and deduce that the

coefficient of this term in g(t) cannot be an integer.

(e) Conclude that, in (a), f(t) is irreducible over Q if and only if

g(t) is irreducible over Z.

(f) Let h(t) b e a p o ly nomial over Z. Show that h(t) is irreducible

over Q if and only if it is irreducible over Z.