T5Q5A GIVE ANSWER IN LATEX Duplication of a cube and quadrat

T5Q5A

GIVE ANSWER IN LATEX

Duplication of a cube and quadrature of a circle are two classical problems of Greek geometry. Let us consider their analogues, namely the problems of \'triplicating a cube\' and \'cubing a ball\'. Triplicating a cube means constructing a cube of three times the volume of a given cube, and cubing a ball means constructing a cube of the same volume as a given ball; in both cases, \'constructing\' refers to using ruler and compasses only. We may also assume that the given cube has unit side length and the given ball has unit radius. Show that a unit cube cannot be triplicated by a construction using ruler and compasses only, following the steps below. Show that if a unit cube can be triplicated using ruler and compasses only, then u = 3 Squareroot 3 is a constructible number Find the minimal polynomial for u over Q, and apply Corollary 2.8 of Chapter 19. Show that a unit ball cannot be by a construction using ruler and compasses only, following the steps below. Show that if a unit ball can be cubed using ruler and compasses only, then upsilon = 3 Squareroot 4 pi/3 is a constructible number. Show that upsilon is transcendental over Q, and apply Theorem 2.7 of Chapter 19.

Solution

We begin with the unit line segment defined by points (0,0) and (1,0) in the plane. We are required to construct a line segment defined by two points separated by a distance of 33. It is easily shown that compass and straightedge constructions would allow such a line segment to be freely moved to touch the origin, parallel with the unit line segment - so equivalently we may consider the task of constructing a line segment from (0,0) to (33, 0), which entails constructing the point (33, 0).

Respectively, the tools of a compass and straightedge allow us to create circles centred on one previously defined point and passing through another, and to create lines passing through two previously defined points. Any newly defined point either arises as the result of the intersection of two such circles, as the intersection of a circle and a line, or as the intersection of two lines. An exercise of elementary analytic geometry shows that in all three cases, both the x- and y-coordinates of the newly defined point satisfy a polynomial of degree no higher than a quadratic, with coefficients that are additions, subtractions, multiplications, and divisions involving the coordinates of the previously defined points (and rational numbers). Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of generated by the previous coordinates. Therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1.

So, given a coordinate of any constructed point, we may proceed inductively backwards through the x- and y-coordinates of the points in the order that they were defined until we reach the original pair of points (0,0) and (1,0). As every field extension has degree 2 or 1, and as the field extension over of the coordinates of the original pair of points is clearly of degree 1, it follows from the tower rule that the degree of the field extension over of any coordinate of a constructed point is a power of 2.

Now, p(x) = x³ 3 = 0 is easily seen to be irreducible over – any factorisation would involve a linear factor (x k) for some k , and so k must be a root of p(x); but also kmust divide 3, that is, k = 1, 3, 1 or 3, and none of these are roots of p(x). By Gauss\'s Lemma, p(x) is also irreducible over , and is thus a minimal polynomial over for 33. The field extension (33): is therefore of degree 3. But this is not a power of 2, so by the above, 33 is not the coordinate of a constructible point, and thus a line segment of 33 cannot be constructed, and the cube cannot be doubled.