460 15. MODERN ALGEBRA15.3. Cauchy's theorem that every cycle of order 3 can be written as the compo-
sition of two cycles of order m if m > 3 looks as if it ought to apply to cycles of
order 2 also. What goes wrong when you try to prove this "theorem"?
15.4. Let Sj (a, 6, c, d) be the jth elementary symmetric polynomial, that is, the sum
of all products of j distinct factors chosen from {a, b, c, d}. Prove that Sj (a, b, c, d) =
Sj(b, c, d) + a5j_i(6, c, d). Derive as a corollary that given a polynomial equationx^4 - Si (a, b, c, d)x^3 + Szla, b, c, d)x^2 + S3 (a, b, c, d)x + (^5) r(a, b, c, d) — 0 = x^4 - fix^3 +
P2X^2 — P3X + PA having a, b,c,d as roots, each elementary symmetric function in
b,c, d can be expressed in terms of a and the coefficients py. S(b,c,d) = p\ — a,
S2(b, c, d) = P2 - aSi(b,c,d) = p 2 - ap\ + a^2 , 53(6,c, d) = p 3 - ap 2 + a^2 p\ - a^3.
15.5. Prove that if æ is a prime in the ring obtained by adjoining the pth roots of
unity to the integers (where ñ is a prime), the equation
zv = xp + yp
can hold only if ÷ = 0 or y — 0.
15.6. Consider the complex numbers of the form æ = ôç + çù, where ù = —1/2 +
\/-3/2 is a cube root of unity. Show that N(z) = m^2 — mn + n^2 has the property
N(zw) = N(z)N{w) and that N(z + w) < 2(N(z) + N(w)). Then show that a
Euclidean algorithm exists for such complex numbers: Given æ and w ö 0, there
exist q and r. Such that æ = qw + r where N(r) < N(w). Thus, a Euclidean
algorithm exists for these numbers, and so they must exhibit unique factorization.
[Hint: N(z) = \z\^2. Show that for every complex number u there exists a number
q of this form such that \q - u\ < 1. Apply this fact with u = z/w and define r to
be æ — qui.}
15.7. Show that in quaternions the equation X^2 + r^2 = 0, where r is a positive
real number (scalar), is satisfied precisely by the quaternions X = ÷ + î such that
÷ — 0, ]î\ = r, that is, by all the points on the sphere of radius r. In other words,
in quaternions the square roots of negative numbers are simply the nonzero vectors
in three-dimensional space. Thus, even though quaternions act "almost" like the
complex numbers, the absence of a commutative law makes a great difference when
polynomial algebra is considered. A linear equation can have only one solution, but
a quadratic equation can have an uncountable infinity of solutions.