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136 4. Equations

express the relationship between the groups G, Gi and HI by the notation:

G1 Y G/HI.

Finally, we subject the base field to a second enlargement by adjoining

2114 to Q(i) to get the field Q(i,2rj4), which is the splitting field of the

quartic polynomial. This field consists of numbers of the form

where a, b, c, d E Q(i).
The group Kr of the quartic polynomial over Q(i, 2114) consists solely
of the identity permutation. For example, any permutation must keep tl
fixed since ti = 1 + 2114 is a polynomial equation over Q(i, 2i14) satisfied
by the roots.
There are two groups of interest: G/H1 and HI, which correspond to
the adjunction of the two radicals i and 2114 to the base field. The main
idea to grasp is that in some sense we can describe the degree of symmetry
displayed by the roots of a polynomial equation by means of a certain group
of permutations, and that this group can be broken down into component
parts through successive adjoinings to the base field for the polynomial. In
general, suppose we have a polynomial p(t) over a base field Fo for which
the equation is solvable by radicais. This means that the splitting field for
the polynomial is the culmination of a sequence of intermediate fields, each
of which is obtained from its predecessor by adjoining all kth roots of some
number a:

where, for each j > I, Fj = Fj-l(ai’“‘) with aj E Fj-1. Compute the
groups of p(t) over these fields, Gj being the group over Fj. Then

where the last group consists of the identity permutation alone. Each field
Fj is the splitting field of a polynomial of the form tk - a over Fj-1; the
group of this polynomial over Fj-1 is Hi. We can write Hj 2 Gj-l/Gi to
indicate that there is a close relationship among the three groups.
It can be shown that the groups Hj arising from adjunctions of radicals
are characterized by a special property, and this in turn imposes a restric-
tion on the structure of G. However, it is possible to find polynomials with
degree as low as 5 whose groups do not satisfy the restriction, and therefore
whose equations cannot be solved by radicals.
What is the group associated with the polynomial t3 - 3t f I (Exercise
3.10)?
E.40. Constructions Using Ruler and Compasses. An early topic
in many Euclidean geometry courses is ruler-and-compasses constructions.
This reflects the ancient Greek interest in the so-called Three Famous Prob-
lems of Antiquity, namely