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132 4. Equations

and quartics of finding the roots of polynomial equations of arbitrary de-
gree. To formulate the type of solution we mean, let us look at polynomials
of low degree.
To find the zero of at + b, we simply divide one coefficient by another;
thus, the roots can be found within the field containing the coefficients by
means of a field operation (division) performed on the coefficients.
The quadratic is more complicated. Finding the zeros often leads us
outside of the smallest field containing the coefficients, but in a special
way-through taking square roots. Thus, with the field operations of ad-
dition, subtraction, multiplication and division, as well as the taking of
square roots, the root of any quadratic is accessible from its coefficients,
and the standard quadratic formula displays this explicitly.
In solving a general cubic equation, we had to first solve a quadratic and
then take cube roots. Thus, we can obtain the roots of a cubic equation
from its coefficients, provided we allow the field operations as well as the
extraction of square and cube roots. Finally, any method for solving a
quartic equation involved the field operations and the extractions of square
and cube roots.
Keeping these cases in mind, we say that a polynomial equation p(t) = 0
is solvable by radicals if and only if the roots of p(t) = 0 are determinable
from the coefficients by means of the field operations and the extraction of
kth roots for certain integers k performed in some order. A radical is any
number of the form cl/‘. There exist polynomials of the fifth and higher
degree which are not solvable by radicals. The analysis of this theorem
requires theory of groups, fields and vector spaces beyond the scope of this
book, but the range of ideas can be indicated by means of an example.
The quartic equation

t4 - 4t3 + 6t2 - 4t - 1 = 0

can be rewritten as (t - 1)” = 2. It has the four roots tl = 1 + 2114,
t2 = 1 - 2114, t3 = 1 + 2114i, t4 = 1 - 21j4i where 2114 denotes the
positive fourth root of 2. None of these roots liis in Q, the smallest field
which contains the coefficients of the polynomial. However, there is a way
of telling how much we have to add to Q in order to get the roots.
We begin with the observation that two rational numbers, a and b, are
distinguishable in the sense that there are polynomial equations over Q
which are satisfied by one but not by the other. Such an equation would
be t - a = 0, which is satisfied by a but not by b. However, & and -&
are indistinguishable in the sense that any polynomial equation over Q
which has one of these numbers as a root must also have the other (try to
disprove this statement). In general, we look at various subsets of the roots
of a polynomial and examine how these can be distinguished from others
by polynomial equations of several variables. Let us see how this works out
in the example before us.