Construction with Straightedge and Compass
We pause in our general development to examine some implications of the results obtained so far in some familiar, geometric situations.
A real number is said to be a constructible number if by the use of straightedge and compass alone we can construct a line segment of length . We assume that we are given some fundamental unit length. Recall that from high-school geometry we can construct with a straightedge and compass a line perpendicular to and a line parallel to a given line through a given point. From this it is an easy exercise (see Problem 1) to prove that if and are constructible numbers then so are , and when . Therefore, the set of constructible numbers form a subfield, , of the field of real numbers.
In particular, since must contain , the field of rational numbers. We wish to study the relation of to the rational field.
Since we shall have many occasions to use the phrase ``construct by straightedge and compass'' (and variants thereof) the words construct, constructible, construction, will always mean by straightedge and compass.
If we can reach from the rational field by a finite number of constructions.
Let be any subfield of the field of real numbers. Consider all the points in the real Euclidean plane both of whose coordinates and are in ; we call the set of these points the plane of . Any straight line joining two points in the plane of has an equation of the form where are all in (see Problem 2). Moreover, any circle having as center a point in the plane of and having as radius an element of has an equation of the form , where all of are in (see Problem 3). We call such lines and circles lines and circles in .
Given two lines in which intersect in the real plane, then their intersection point is a point in the plane of (see Problem 4). On the other hand, the intersection of a line in and a circle in need not yield a point in the plane of . But, using the fact that the equation of a line in is of the form and that of a circle in is of the form , where are all in , we can show that when a line and circle of intersect in the real plane, they intersect either in a point in the plane of or in the plane of for some positive in (see Problem 5). Finally, the intersection of two circles in can be realized as that of a line in and a circle in , for if these two circles are
then their intersection is the intersection of either of these with the line
so also yields a point either in the plane of or of for some positive in .
Thus lines and circles of lead us to points either in or in quadratic extensions of . If we now are in for some quadratic extension of , then lines and circles in intersect in points in the plane of where is a positive number in . A point is constructible from if we can find real numbers , such that
such that the point is in the plane of Conversely, if is such that is real then we can realize as an intersection of lines and circles in (see Problem 6). Thus a point is constructible from if and only if we can find a finite number of real numbers , such that
(1) or 2;
(2) or 2 for ;
and such that our point lies in the plane of .
We have defined a real number to be constructible if by use of straightedge and compass we can construct a line segment of length . But this translates, in terms of the discussion above, into: is constructible if starting from the plane of the rational numbers, , we can imbed in a field obtained from by a finite number of quadratic extensions. This is
THEOREM 5.4.1: The real number is constructible if and only if we can find a finite number of real numbers such that
(2) for ,
such that .
However, we can compute the degree of over , for by Theorem 5.1.1
Since each term in the product is either 1 or 2, we get that
and thus the
COROLLARY 1: If is constructible then lies in some extension of the rationals of degree a power of 2.
If is constructible, by Corollary 1 above, there is a subfield of the real field such that and such that . However, , whence by the corollary to Theorem 5.1.1
thereby is also a power of 2. However, if satisfies an irreducible polynomial of degree over , we have proved in Theorem 5.1.3 that . Thus we get the important criterion for nonconstructibility
COROLLARY 2: If the real number satisfies an irreducible polynomial over the field of rational numbers of degree , and if is not a power of 2, then is not constructible.
This last corollary enables us to settle the ancient problem of trisecting an angle by straightedge and compass, for we prove
THEOREM 5.4.2: It is impossible, by straightedge and compass alone, to trisect .
Proof: If we could trisect by straightedge and compass, then the length would be constructible. At this point, let us recall the identity
Putting and remembering that we can obtain
Thus is a root of the polynomial over the rational field. However, this polynomial is irreducible over the rational field (Problem , and since its degree is 3, which certainly is not a power of 2, by Corollary 2 to Theorem 5.4.1, is not constructible. Thus cannot be trisected by straightedge and compass.
Another ancient problem is that of duplicating the cube, that is, of constructing a cube whose volume is twice that of a given cube. If the original cube is the unit cube, this entails constructing a length such that . Since the polynomial is irreducible over the rationals (Problem 7(b)), by Corollary 2 to Theorem 5.4.1, is not constructible. Thus
THEOREM 5.4.3: By straightedge and compass it is impossible to duplicate the cube.
We wish to exhibit yet another geometric figure which cannot be constructed by straightedge and compass, namely, the regular septagon. To carry out such a construction would require the constructibility of . However, we claim that satisfies (Problem 8) and that this polynomial is irreducible over the field of rational numbers (Problem ). Thus again using Corollary 2 to Theorem 5.4.1 we obtain
THEOREM 5.4.4: It is impossible to construct a regular septagon by straightedge and compass.