Editing Field (mathematics) (section)

=== Constructible numbers ===
[[File:Root_construction_geometric_mean5.svg|thumb|255px|The [[geometric mean theorem]] asserts that {{math|1=''h''<sup>2</sup> = ''pq''}}. Choosing {{math|1=''q'' = 1}} allows construction of the square root of a given constructible number {{math|''p''}}.]]
{{main|Constructible numbers}}

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with [[compass and straightedge]]. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of [[constructible numbers]].<ref>{{harvp|Artin|1991|loc=Chapter 13.4}}</ref> Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only [[Compass (drawing tool)|compass]] and [[straightedge]]. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field {{math|'''Q'''}} of rational numbers. The illustration shows the construction of [[square root]]s of constructible numbers, not necessarily contained within {{math|'''Q'''}}. Using the labeling in the illustration, construct the segments {{math|''AB''}}, {{math|''BD''}}, and a [[semicircle]] over {{math|''AD''}} (center at the [[midpoint]] {{math|''C''}}), which intersects the [[perpendicular]] line through {{math|''B''}} in a point {{math|''F''}}, at a distance of exactly <math>h=\sqrt p</math> from {{math|''B''}} when {{math|''BD''}} has length one.

Not all real numbers are constructible. It can be shown that <math>\sqrt[3] 2</math> is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a [[Doubling the cube|cube with volume 2]], another problem posed by the ancient Greeks.