Problems of geometric construction appeared early in the history of mathematics. They were first considered by the Greek mathematicians of the fifth century B.C. Only two instruments—an unmarked ruler and a compass—were permitted in these constructions. Although many such constructions could be performed, others eluded the efforts of these mathematicians. Four famous problems from the period that remained unsolved for a long time are the following: doubling the cube, which consists of constructing a cube whose volume is twice that of a given cube; trisecting the angle; squaring the circle, which consists of constructing a square whose area is that of a given circle; and constructing regular polygons.

At the end of the eighteenth century, when it was observed that questions on geometric constructions can be translated into questions on fields, a breakthrough finally occurred. The 19-year-old Gauss [2: art. 365] proved in 1796 that the regular 17-sided polygon is constructible. A few years later, Gauss [2: art. 365, 366] stated necessary and sufficient conditions for the constructibility of the regular n-sided polygon. He gave a proof only of the sufficiency, and claimed to have a proof of the necessity; the latter was first given by Wantzel [1] in 1837. In his investigations, Gauss introduced and used a number of concepts that became of central importance in subsequent developments.