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104 3. Factors and Zeros

(c) Let p be a prime and let C be a primitive pth root of unity.

Define

u = C([” : a is a quadratic nonresidue modulo p, 12 a < p}

v = c{p : a is a quadratic residue modulo p, 1 5 a < p}.

For the cases p = 11, 13, 17, show that u and v are the zeros

of a quadratic polynomial of the form t2 + t + 6, where k is an

integer.

(This has significance for ruler and compasses constructions in the

plane. Since such a construction locates points as intersections of

straight lines and circles, the coordinates of constructible points are

found by solving linear and quadratic equations whose coefficients

belong to the smallest field containing the coordinates of points al-

ready given or constructed. The result of this exercise implies that

the points u and v are constructible in the complex plane once the

unit circle is given. In the case p = 17, it can be shown that one

can solve a succession of quadratic equations, each with coefficients

expressible in terms of the roots of its predecessors, until one finally

obtains the root Ciz itself. As a consequence there exists a ruler-and-

compasses construction for a regular 17-gon in the plane. This result,

due to Gauss, holds when 17 is replaced by any prime, such as 257,

which is 1 plus a power of 2. It is this result which accounts for the

original interest in cyclotomic polynomials.)

19. (a) Show that the coefficient of xn in the expansion of j(x) =
    (x6 + x5 + x4 + x3 + x2 + z)” is the number of ways of rolling a
    total of n with two distinguishable ordinary cubical dice.
    The remainder of the exercise is devoted to assigning numbers to the
    faces of the two dice in such a way that the number of ways of rolling
    a total of n is the same as for two ordinary dice.

(b) Verify the factorization

f(x) = x2(x + 1)2(x2 + x + 1)2(t2 - x + 1)“.

(c) We wish to write f(x) = g(x)h(x) where g(x) # h(x) and

g(x) = xyx + 1)“(22 + x + 1)“(22 - x + l)d

with 0 5 a, b, c, d 2 2. If we can arrange that g(x) and h(s) are

each equal to the sum of six not necessarily distinct terms of the

form xk, then the numbers k can be used to label the faces of

the dice. Argue that, for the desired labeling, we must have that

g(1) = h(1) = 6 and a = b = c = 1. Determine g(x) and h(x).