Abstract Algebra Theory and Applications - Computer Science ...

19.3 GEOMETRIC CONSTRUCTIONS 341Corollary 19.26 The field of all constructible numbers is an algebraic extensionof Q.As we can see by the field of constructible numbers, not every algebraicextension of a field is a finite extension.Doubling the Cube and Squaring the CircleWe are now ready to investigate the classical problems of doubling the cubeand squaring the circle. We can use the field of constructible numbers toshow exactly when a particular geometric construction can be accomplished.Doubling the cube is impossible. Given the edge of the cube, it is impossibleto construct with a straightedge and compass the edge of the cubethat has twice the volume of the original cube. Let the original cube have anedge of length 1 and, therefore, a volume of 1. If we could construct a cubehaving a volume of 2, then this new cube would have an edge of length 3√ 2.However,3√2 is a zero of the irreducible polynomial x 3 − 2 over Q; hence,[Q( 3√ 2 ) : Q] = 3This is impossible, since 3 is not a power of 2.Squaring the circle is impossible. Suppose that we have a circle of radius1. The area of the circle is π; therefore, we must be able to construct asquare with side √ π. This is impossible since π and consequently √ π areboth transcendental. Therefore, using a straightedge and compass, it is notpossible to construct a square with the same area as the circle.Trisecting an AngleTrisecting an arbitrary angle is impossible. We will show that it is impossibleto construct a 20 ◦ angle. Consequently, a 60 ◦ angle cannot be trisected. Wefirst need to calculate the triple-angle formula for the cosine:cos 3θ = cos(2θ + θ)= cos 2θ cos θ − sin 2θ sin θ= (2 cos 2 θ − 1) cos θ − 2 sin 2 θ cos θ= (2 cos 2 θ − 1) cos θ − 2(1 − cos 2 θ) cos θ= 4 cos 3 θ − 3 cos θ.The angle θ can be constructed if and only if α = cos θ is constructible. Letθ = 20 ◦ . Then cos 3θ = cos 60 ◦ = 1/2. By the triple-angle formula for the