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3.1. EQUATIONS IN ONE VARIABLE 59
In fact, 2 if we choose such a polynomial “at random”, the probability of its
having a solution that can be expressed in terms of radicals is zero. Of course,
any particular quintic, or higher degree equation, may have solutions expressible
in radicals, such as x 5 − 2, whose solutions are 5√ 2, but this is the exception
rather than the rule.
Hence algebra systems, if they handle such concepts, can only regard the
roots of such equations as being defined by the polynomial of which they are
a root. A Maple example 3 is given in figure 1.5.1, where the Maple operator
RootOf is generated. It is normal to insist that the argument to RootOf (or its
equivalent) is square-free: then different rooots are genuinely different. Then α,
the first root of f(x), satisfies f(α) = 0, the second root β satisfies f(x)/(x−α) =
0, and so on. Even if f is irreducible, these later polynomials may not be, but
determining the factorisations if they exist is a piece of Galois theory which
would take us too far out of our way [FM89]. It is, however, comparatively easy
to determine the Monte Carlo question: “such factorisations definitely do not
exist”/“they probably do exist” [DS00].
3.1.5 Reducible defining polynomials
It should be noted that handling such constructs when the defining polynomial
is not irreducible can give rise to unexpected results. For example, in Maple,
1
if α is RootOf(x^2-1,x), then
α−1
returns that, but attempting to evaluate
this numerically gives infinity, which is right if α = 1, but wrong if α = −1,
the other, equally valid, root of x 2 − 1. In this case, the mathematical answer
to “is α − 1 zero?” is neither ‘yes’ nor ‘no’, but rather ‘it depends which α
you mean’, and Maple is choosing the 1 value (as we can see from 1
α+1 , which
evaluates to 0.5). However, the ability to use polynomials not guaranteed to
be irreducible can be useful in some cases — see section 3.3.7. In particular,
algorithm 10 asks if certain expressions are invertible, and a ‘no’ answer here
entrains a splitting into cases, just as asking “is α−1 zero?” entrains a splitting
of RootOf(x^2-1,x).
In general, suppose we are asking if g(α) is invertible, where α = RootOf(f(x), x),
i.e. we are asking for d(α) such that d(α)g(α) = 1 after taking account of the
fact that α = RootOf(f(x), x). This is tantamount to asking for d(x) such that
d(x)g(x) = 1 modulo f(x) = 0, i.e. d(x)g(x) + c(x)f(x) = 1 for some c(x). But
applying the Extended Euclidean Algorithm (Algorithm 4) to f and g gives us
c and d such that cf + dg = gcd(f, g). Hence if the gcd is in fact 1, g(α) is
invertible, and we have found the inverse.
If in fact the gcd is not 1, say some h(x) ≠ 1, then we have split f as f = hĥ,
2 The precise statement is as follows. For all n ≥ 5, the fraction of polynomials of degree n
and coefficients at most H which have a root expressible in radicals tends to zero as H tends
to infinity.
3 By default, Maple will also use this formulation for roots of most quartics, and the expression
in figure 3.2 is obtained by convert(%,radical) and then locating the sub-expressions by
hand. This can be seen as an application of Carette’s view of simplification (page 21), though
historically Carette’s paper is a retrospective justification.