294 Answers to Exercises and Solutions to Problems
Since 0 # 1, it is the second factor which vanishes and we obtain
1 = l-e+e2--3+...+em-l_,,
= (i-e)(i+e2+...+em-l).Hence,
(I- e)-l = i + e2 + e4 +... + em--l = e + e3 + e5 +... + em-2.8.19. (a) If p(x) = x2 + bx + c, q(x) = x2 + dx + e, choose h and k such
that2h+b=dandh2+bh+c-k=e.
(b) The reducible manic quadratics are of the form (x - r)(x - s), where
r, s E F; there are m(m+ 1)/2 of these. The total number of manic quadrat-
ics is m2, so there are m(m - 1)/2 irreducible quadratics. From (a), it can
be seen that for each of the m choices of linear coefficient, the number
of irreducible quadratics is the same. Hence, for a given choice, there are
(m - 1)/2 irreducible quadratics for appropriate values of k.
8.20. If the four terms are a - d, a, a + d, a + 2d, we obtain
(a - d)a(a + d)(a + 2d) + 8 = (a” + ad - d2)2.If a2 + ad - d2 = b2, then (2a + d)2 = 4b2 + 5d2. For example, b = 1,
d = 3 leads to a fourth power. A more general solution appears in Amer.
Math. Monthly 57 (1950) 186.
8.21. (b) Suppose f = uu .. .uik, where the ui are irreducible with
any pair coprime and ai 1 1, and g = ui1u2 ... up, where bi 2 0. Since
f = g”‘h with g not dividing h, it follows that ai 2 mbi for each i and that,
for some irreducible factor, say ui, ai < (m+ l)bl. Now UT’-’ but not UT’
divides f'. Since al - 1 < (m + l)bl - 1 < (m + l)bl, it follows that gm+’
does not divide f’.
8.22. By de Moivre’s Theorem (Exercise 1.3.8), cosnB+ isinn = u(cos0,
sin2 0) + i sin Bv(cos 0, sin2 f?), for some polynomials u and v. Let x = cos 0.
Then the result holds with f(x) = T,(x) = cosne and g(x) = v(x, 1 -x2).
8.23. Since xk - 1 = H{Qd(x) : dlk}, {m}! is a product of factors
&d(x), where each Qd(x) occurs as often as d divides a number in the
set {1,2,..., m}; the exponent of &d(z) is [m/dJ. It suffices to show that,
for each positive integer d,
[(m + n)/dJ + [m/&J + 144 I 1244 + 1244 a
Let m = ud+r, n = vd +s where 0 5 r, s < d. Then the difference between
the right and left sides is
12r/dJ + 12s/dJ - [(r + s)/dJ.