In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
The ancient Greek mathematicians first attempted straightedge-and-compass constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and regular polygons of 3, 4, or 5 sides or one with twice the number of sides of a given polygon. But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or regular polygons with other numbers of sides. Nor could they construct the side of a cube whose volume is twice the volume of a cube with a given side.
Squaring the circle
The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass.
Squaring the circle has been proved impossible, as it involves generating a transcendental number, that is, √π. Only certain algebraic numbers can be constructed with ruler and compass alone, namely those constructed from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.
Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity.
Doubling the cube
Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over the rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass.
Angle trisection
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2π/5 radians (72° = 360°/5) can be trisected, but the angle of π/3 radians (60°) cannot be trisected. The general trisection problem is also easily solved when a straightedge with two marks on it is allowed which is called a neusis construction.
Distance to an ellipse
The line segment from any point in the plane to the nearest point on a circle can be constructed, but the segment from any point in the plane to the nearest point on an ellipse of positive eccentricity cannot in general be constructed.
Alhazen's problem
In 1997, the Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient Alhazen's problem which is also called the billiard problem or reflection from a spherical mirror.
Constructing regular polygons
Some regular polygons, say a pentagon, are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass?
Carl Friedrich Gauss in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular n-sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary; the conjecture was proven by Pierre Wantzel in 1837.
The first few constructible regular polygons have the following numbers of sides:
3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272. There are known to be an infinitude of constructible regular polygons with an even number of sides. Because if a regular n-gon is constructible, then so is a regular 2n-gon and hence a regular 4n-gon, 8n-gon, etc. However, there are only 31 known constructible regular n-gons with an odd number of sides.
Galois Field theory is applied to characterize ruler-and-compass constructions and is applied in geometry to identify construcatable shapes.
QUESTION I: How Galois Field Theory is used to prove the unsolvability of general quintic or higher order polynomials?
QUESTION II: Could constructabilities be studied in non-euclidean spaces?
REFERENCE: Wikipedia