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§ 6.3 Geometric constructions

 
Definition 6.3.1. The real number a is said to be a constructible number if it is possible to construct a line segment of length |a| by using only a straightedge and compass.

 
Proposition 6.3.2. The set of all constructible real numbers is a subfield of the field of all real numbers.

 
Definition 6.3.3. Let F be a subfield of R. The set of all points (x,y) in the Euclidean plane R² such that x,y belong to F is called the plane of F.
    A straight line with an equation of the form ax + by + c = 0, for elements a,b,c in F, is called a line in F.
    Any circle with an equation of the form x² + y² + ax + by + c = 0, for elements a,b,c in F, is called a circle in F.

 
Lemma 6.3.4. Let F be a subfield of R.

(a) Any straight line joining two points in the plane of F is a line in F.

(b) Any circle with its radius in F and its center in the plane of F is a circle in F.

 
Lemma 6.3.5. The points of intersection of lines in F and circles in F lie in the plane of F(u), for some u in F.

 
Theorem 6.3.6. The real number u is constructible if and only if there exists a finite set u₁, u₂, . . . , u_n of real numbers such that

(i) u₁² belongs to Q,

(ii) u_i² belongs to Q(u₁,...,u_i-1), for i=2,..., and

(iii) u belongs to Q(u₁,...,u_n).

 
Corollary 6.3.7. If u is a constructible real number, then u is algebraic over Q, and the degree of its minimal polynomial over Q is a power of 2.

 
Theorem 6.3.9. It is impossible to find a general construction for trisecting an angle, duplicating a cube, or squaring a circle.

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