Answers to Exercises; Chapter 3 281
On the other hand, if-c is a primitive Pkth root of unity, then (-0” =-1, from which < k = (^1). It is straightforward to see that C is a primitive
kth root of unity. Hence 6 is a primitive kth root of unity iff -(’ is a
primitive 2kth root of unity. Thus t&(t) = Qk(-t), since deg&k(t) is
even by Exercise 8.
5.13. cos[(2k + l)a/n] + isin[(2k+l)?r/n],fork=O,l,..., n-l.
5.14. r’l”(cos 2ka/n + i sin 2k?r/n) (0 5 k < n - 1).
5.15. Let n = kr. Then, if u = tr, we have t” - 1 = uk - 1 = (u - 1)
(uk-’ +... + l), from which the result follows. The conjecture is false; take
k = 1, m = 2.
5.17. Use C + C2 +.. f + C6 = -1 and C7 = 1 to check that u+ v = -1,
uv= 2.
5.18. (a) Modulo 11: 1, 3, 4, 5,9; Modulo 13: 1, 3,4, 9, 10, 12; Modulo 17:
1, 2, 4, 8, 9, 13, 15, 16.
(b)Foranyprimep,u+v=C{C” : l<u<p-l}=-l.Whenp=ll,
13, 17, the product uv = 3, -3, -4, respectively.
5.19. f(x) = (x4 + 2x3 + 2x2 + x)(xa + x6 + x5 + x4 + x3 + x). The faces
of the dice are labeled (4,3,3,2,2,1) and (8,6,5,4,3,1). For more on this
consult :
Duane Broline, Renumbering the faces of dice. Math. Mug. 52 (1979),
312-315.
J.A. Gallian & D.J. Rusin, Cyclotomic polynomials and nonstandard
dice. Discrete Math. 27 (1979), 245-249.
Martin Gardner, Mathematical games. Scientific American 238 (1978),
19-32.
6.1. Suppose f = u/v. Divide v into ti to obtain 2~ = pv + w where deg w <
degv. Then the result follows with g = w/v.
6.2. Putting the sum A/(t -m)+ B/(t - n ) over a common denominator and
equating the numerator to at + b yields the condition A(t - n) + B(t - m) =
at + b for t # m, n. This can be interpreted as saying that the polynomial
A(t - n) + B(t - m) - (at + b) vanishes for infinitely many values oft (all
but t = m, n). But this implies that it must be the zero polynomial (by
Exercise 2.2.4) and so vanishes for t = m and t = n.
A = (am + b)/(m - n); B = (an + b)/(n - m).
6.3. A = 3, B = -2.
6.4. (a) The result can be proved by induction on k. It is clearly true when
k = 1. Suppose it has been established when the degree of the denominator
is less than k.