작도 가능성과 원분확대체에 대하여

Abstract

본 논문에서는 작도 가능성 여부를 따지는 데 필요한 체 이론을 도입하여 작도 문제가 대수적인 문제로 전환될 수 있음을 보인 후에 작도가능한 수와 작도 불가능한 수에 대해 알아본다. 이것을 바탕으로 작도가능성의 기준을 밝혀 3대 작도 불능 문제를 증명한 뒤 갈로아 이론과 원분 확대체의 이론을 바탕으로 작도 가능하기 위한 필요충분조건을 살펴본다. 그 후 마지막으로 정17각형을 작도하는데 필요한 이론적인 배경을 설명하고, 실제로 정17각형의 작도법을 소개한다.; Greek mathematicians as far back as the fourth century B.C. had tried without success to find geometric constructions using straightedge and compass to trisect the angle, double the cube, and square the circle. Although they were never able to prove that such constructions were impossible, they did manage to construct the solutions to these problems using other tools, including the conic sections. It was Carl Gauss in the early nineteenth century who made a detailed study of constructibility in connection with his solution of cyclotomic equations, the equations of the from x^(p)=0 with p prime whose roots form the vertices of a regular p-gon. He showed that although all such equations are solvable using radicals, if p-1 is not a power of 2 then the solutions must involve roots higher than the second. In fact, Gauss asserted that anyone who attempted to find a geometric construction for a p-gon where p-1 is not a power of 2 would "spend his time uselessly." Interestingly, Gauss did not prove the assertion that such construction were impossible. That was accomplished in 1837 by Pierre Wantzel (1814-1848), who in fact proved 'If γis constructible and , γ?Q, then there is a finite sequence of real numbers α₁,…,α_(n)=γ such that Q(α₁, …, α_(i)) is an extension of Q(α,…,α_(i-1)) of degree 2. In particular, [Q(γ):Q]=2^(γ) for some integer r≥0 .' In Euclidean geometry, a ruler is only used to draw a straight line through two given points and a compass, together with the help of a ruler, is used to draw a circle with center at a given point and radius equal to the distance between two given points. The standard of a construction in which ruler and compass are used is as follows. 1. We can draw a straight line which passes by the given two points. 2. We can construct an intersection points of two distinct straight lines. 3. We can construct an intersection point df a straight line and a circle. 4. We can construct an intersection point of two distinct circles. From these basic principles, the coordinates of all points which can be constructed on a coordinate plane are expressed by addition, subtraction, multiplication, division and square root of integers. In this article, We review the constructibility of regular n-gons. In particular, we study the theoretical background for constructing a regular 17-gon.