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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**.

**(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
u_{1},
u_{2},
. . . ,
u_{n}
of real numbers such that

**(i)**u_{1}^{2}belongs to**Q**,**(ii)**u_{i}^{2}belongs to**Q**(u_{1},...,u_{i-1}), for i=2,..., and**(iii)**u belongs to**Q**(u_{1},...,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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