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 \sqrt[3]{2}. 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 (\sqrt[3]{2}, 0), which entails constructing the point (\sqrt[3]{2}, 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 - and -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 - and -coordinates have
minimal polynomials of degree at most 2 over the
subfield of
\mathbb{R} 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 - and -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
\mathbb{Q} 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 \mathbb{Q} of any coordinate of a constructed point is a
power of 2. Now, is easily seen to be
irreducible over
\mathbb{Z} – any
factorisation would involve a
linear factor for some k \isin \mathbb{Z}, and so must be a
root of ; but also must divide 2 (by the
rational root theorem); that is, or , and none of these are roots of . By
Gauss's Lemma, is also irreducible over \mathbb{Q}, and is thus a minimal polynomial over \mathbb{Q} for \sqrt[3]{2}. The field extension \mathbb{Q} (\sqrt[3]{2}):\mathbb{Q} is therefore of degree 3. But this is not a power of 2, so by the above: • \sqrt[3]{2} is not the coordinate of a constructible point, so • a line segment of \sqrt[3]{2} cannot be constructed with ruler and compass, and • the cube cannot be doubled using only a ruler and a compass. ==History==