( Note that by designating cardinal directions for 1 ,-1,, and, complex numbers such as are considered constructible.

The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension of which is closed under square root and complex conjugation.

Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients.

In particular, any constructible point ( or length ) is an algebraic number, though not every algebraic number is constructible ( i. e. the relationship between constructible lengths and algebraic numbers is not bijective ); for example, is algebraic but not constructible.

The angles that are constructible form an abelian group under addition modulo 2π ( which corresponds to multiplication of the points on the unit circle viewed as complex numbers ).

Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

In abstract terms, using these more powerful tools of either neusis using a markable ruler or the constructions of origami extends the field of constructible numbers to a larger subfield of the complex numbers, which contains not only the square root, but also the cube roots, of every element.

The numbers that can be constructed are called the origami or pythagorean numbers, if the distance between the two given points is 1 then the constructible points are all of the form where and are Pythagorean numbers.

Adding the fifth axiom gives the Euclidean numbers, that is the points constructible by straightedge and compass constructions.

Then 0 < sup >#</ sup > is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with c < sub > i </ sub > interpreted as the uncountable cardinal ℵ < sub > i </ sub >.

Since there are 5 known Fermat primes, we know of 31 numbers that are multiples of distinct Fermat primes, and hence 31 constructible odd-sided regular polygons.

It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible.

There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle.

The angles that are constructible are exactly those whose tangent ( or equivalently, sine or cosine ) is constructible as a number.

For example the real part, imaginary part and modulus of a point or ratio z ( taking one of the two viewpoints above ) are constructible as these may be expressed as

It is open whether there are infinitely many primes p for which a regular p-gon is constructible with ruler, compass and an angle trisector.

: Which regular polygons are constructible polygons?

* Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer ; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not.

* The axiom of constructibility ( which asserts that all sets are constructible );

The Whitehead conjecture is true if all sets are constructible.

In particular, the constructible regular polygons with these axioms are those with sides, where is a product of distinct Pierpont primes.

By contrast, in Gödel's constructible universe L, one uses only those subsets of the previous stage that are:

A point in the Euclidean plane is a constructible point if, given a fixed coordinate system ( or a fixed line segment of unit length ), the point can be constructed with unruled straightedge and compass.

It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length | r | can be constructed with compass and straightedge.

By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions.

* a regular polygon that can be constructed with compass and straightedge ; see constructible polygon.

One can show that any number constructible in one step from a field is a solution of a polynomial of degree 2, and therefore any number which is constructible by a series of steps is the solution of a minimal polynomial whose degree is a power of 2.

For example, for any constructible angle, the angle can be trivially trisected by ignoring the given angle and directly constructing an angle of measure.

The regular heptadecagon is a constructible polygon ( that is, one that can be constructed using a compass and unmarked straightedge ), as was shown by Carl Friedrich Gauss in 1796 at the age of 19.

A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N ( x )< sup > β / 2 </ sup >, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β.

In mathematics, the constructible universe ( or Gödel's constructible universe ), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets.

In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC ( and even in much weaker theories ).

It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant.

The regular icosagon is a constructible polygon, by an edge-bisection of a regular decagon, and can be seen as a truncated decagon.

Every point constructible using straightedge and compass may be constructed using compass alone.

The field extension generated by any additional point constructible in this larger field has degree a multiple of a power of two and a power of three, and may be broken into a tower of extensions of degree 2 and 3.

* a set in Kurt Gödel's universe L, which may be constructed by transfinite application of certain constructions in set theory ; see constructible universe.

The problem is known to be impossible to solve with only compass and straightedge, because ≈ 1. 25992105 is not a constructible number.

But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible ; that is, it cannot be constructed with straightedge and compass.

This is close to being best possible, because the existence of 0 < sup >#</ sup > implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0 < sup >#</ sup >.

Since the initial Adleman experiments, advances have been made and various Turing machines have been proven to be constructible.