102 3. Factors and Zeros
- By a suitable pairing of linear factors over C of the polynomial P,(t),
show that we can obtain a factorization of P,(t) as a product of
irreducible polynomials over R as follows:
(n-1)/2
for n odd, P,(t) = (t - 1) n (t2 - 2cos(2kn/n)t + 1)
k=l(n-2)/2
for n even, P,(t) = (t - l)(t + 1) n (t2 - 2cos(2klr/n)t + 1)
k=l
In particular, obtain a factorization of Pa(t) and P5(t) over R.- Show that, among the 8 distinct 8th roots of 1, there are
(i) one square root of 1 other than 1 itself;
(ii) two fourth roots of 1 which are not square roots of 1;
(iii) four eighth roots of 1 which are not fourth roots of 1.- Which 12th roots of 1 are roots of lower degree? Make a table showing
each 12th root of 1 and the minimum exponent to which it must be
raised to yield 1. - A complex number C is a primitive nth root of unity if and only if
C” = 1 but Ck # 1 for each integer k with 1 5 k 5 n - 1. That is, n
is the smallest exponent to which C can be raised to yield 1:
(a) Verify that there are 4 primitive 8th roots and 4 primitive 12th
roots of unity.
(b) Let w be an nth root of unity. Show that there exists a positive
number m such that (i) m 1 n and (ii) w is a primitive mth root
of unity.- For positive integer n, let <,, = cos(2a/n) + isin(2n/n).
(a) Show that the zeros of Pn(t) are precisely the powers of <“.
(b) Show that 1 + Cn + <i +... + C-’ = 0.
(c) Prove that the primitive nth roots of 1 are the numbers <z where
l<a<n- 1 and gcd(a,n) = 1. (Test this for specific n, such
as n = 12.)
(d) For p a prime, show that every pth root of unity is primitive
except 1 itself.
(e) Show that, for n 2 3, the number of primitive nth roots of unity
is even.