Constructible number
We say that a real number is constructible if a segment of length
can be constructed with a straight edge and compass starting with a segment of length
.
We say that a complex number is constructible if
and
are both constructible (we also say that the point
is constructible). It is easy to show that
is constructible iff the point
can be constructed with a straight edge and compass in the cartesian plane starting with the points
and
. (Notice that our two definitions coincide when
is a real number.)
Characterization Theorem
It is possible to completely characterize the set of all constructible numbers:
A complex number is constructible iff it can be formed from the rational numbers in a finite number of steps using only the operations addition, subtraction, multiplication, division, and taking square roots.
For instance, this means one can construct segments of length: and
, but one cannot construct a segment of length
.
This condition can be rephrased in terms of field theory as follows:
A complex number is constructible iff there is a chain of field extensions
such that each extension
is quadratic (i.e.
).
This is equivalent because the field extension is quadratic iff
for some
with
, so taking a square root in the above construction is equivalent to taking at most a quadratic extension of a field, while adding, subtracting, multiplying or dividing does not add anything to the field. (Does someone else want to phrase that better?)
Using this second characterization (and the tower law) we get the necessary (but not sufficient) condition that for some nonnegative integer
, or equivalently that
is algebraic and it's minimal polynomial has degree
.
Using this theorem one can easily answer many classical construction problems, such as the three Greek problems of antiquity and the question of which regular polygons are constructible.