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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 R^{2}
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
x^{2} + y^{2} + 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.
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
u_{1},
u_{2},
. . . ,
u_{n}
of real numbers such that
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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