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4.3. Solving Special Polynomial Equations 131

(a) Solve the equation f(t) =^0 by Cardan’s Method (Exercise

1.4.4). To find u3 and v3, we need access to the field Q(n)

obtained by adjoining a to Q, and finally t = 21 + v is

found in the field obtained by further adjoining a cube root

of (l/2)(-1+,&3). Sh ow that the roots of the equation as well

as fl are all contained in the smallest field which contains Q

along with C, a primitive ninth root of unity. Denote this field

by Q(C).

(b) Verify that Cc’ + C3 + 1 = 0 and prove that Q((‘) is an integral

domain. In fact, Q(C) is a field, the hard part to show being that

it is closed under the taking of reciprocals. The general argument

for doing this runs as follows: Let g(t) be any polynomial of

degree less than 6 over Q (so that g(C) is a typical element of

Q(c)). Since t6+t3+1 is irreducible over Q, the greatest common

divisor of g(t) and t6 + t3 + 1 is 1. By the Euclidean algorithm,

we can find polynomials u(t) and v(t) over Q such that

u(t)(P + t3 + 1) + v(t)g(t) = 1.

Set t = C to obtain v(C)g(C) = 1. Use this technique to determine

(‘-l and (c3 + <)-l as polynomials of 6.

(c) With C denoting a primitive ninth root of unity, verify that the

zeros of f(t) are <+Cs, C2+c7, c4+c5 by (i) direct substitution,

(ii) showing that the coefficients of f(t) are suitable symmetric

functions of the zeros.

(d) The field Q(C) contains nonreal numbers. However, all of the ze-

ros of f(t) are real. Argue that the smallest field which contains

Q along with the zeros of f(t) is contained in R and is thus not

Q(C).

(e) Show that, if ‘u is any zero of f(t), then the other two zeros are

u2 - 2 and 2 - ?J - u2. Deduce that

Q(u)={a+bu+cu2 : a,b,cEQ}

is the smallest field containing Q and the zeros of f(t).

Explorations

E.39. Solving by Radicals. Perhaps it is surprising to be told that a

certain mathematical procedure is impossible, that it can never be found

regardless of the time and energy expended in the search. Yet, it can be

shown beyond any doubt, that, because of the underlying structure of the

number system, there is no general method like those for quadratics, cubits