A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x- and y-coordinate axes. A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... Look up one in Wiktionary, the free dictionary. ... In general English usage, length (symbols: l, L) is one particular instance of distance: an objects length is its extent along its longest dimension. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...

It can then be shown that a real number is constructible if and only if, given a line segment of unit length, one can construct a line segment of length | r | with compass and straightedge. It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... Look up one in Wiktionary, the free dictionary. ... In general English usage, length (symbols: l, L) is one particular instance of distance: an objects length is its extent along its longest dimension. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

The set of constructible numbers can be completely characterized in the language of field theory. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack. In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...

Geometric definitions

The geometric definition of a constructible point is as follows. First, for any two distinct points P and Q in the plane, let L(P, Q) denote the unique line through P and Q, and let C(P, Q) denote the unique circle with center P, passing through Q. (Note that the order of P and Q matters for the circle.) By convention, L(P, P) = C(P, P) = {P}. Then a point Z is constructible from E, F, G and H if either

1. Z is in the intersection of L(E, F) and L(G, H), where L(E, F) ≠ L(G, H);
2. Z is in the intersection of C(E, F) and C(G, H), where C(E, F) ≠ C(G, H);
3. Z is in the intersection of L(E, F) and C(G, H).

Since the order of E, F, G, and H in the above definition is irrelevant, the four letters may be permuted in any way. Put simply, Z is constructible from E, F, G and H if it lies in the intersection of any two distinct lines, or of any two distinct circles, or of a line and a circle, where these lines and/or circles can be determined by E, F, G, and H, in the above sense. The term intersection can mean: a road junction, where two roads intersect each other, such as a roundabout intersection; in mathematics, the set in which two or more other sets intersect each other; see intersection (set theory); a movie; see Intersection (movie). ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

Now, let A and A' be any two distinct fixed points in the plane. A point Z is constructible if either

1. Z = A;
2. Z = A'
3. there exist points P₁, ..., P_n, with Z = P_n, such that for all j ≥ 1, P_{j + 1} is constructible from points in the set {A, A', P₁, ..., P_j}.

Put simply, Z is constructible if it is either A or A', or if it is obtainable from a finite sequence of points starting with A and A', where each new point is constructible from previous points in the sequence.

The origin O is defined as follows. The circles C(A, A') and C(A', A) intersect in two distinct points; these points determine a unique line, and the origin O is defined to be the intersection of this line with L(A, A').

Transformation into algebra

All rational numbers are constructible, and all constructible numbers are algebraic numbers. Also, if a and b are constructible numbers with b ≠ 0, then a − b and a/b are constructible. Thus, the set K of all constructible complex numbers forms a field, a subfield of the field of algebraic numbers. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an algebraic number is any number that is a root of a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...

Furthermore, K is closed under square roots and complex conjugation. These facts can be used to characterize the field of constructible numbers, because, in essence, the equations defining lines and circles are no worse than quadratic. The characterization is the following: a complex number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field Q. More precisely, z is constructible if and only if there exists a tower of fields In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... It has been suggested that this article or section be merged with Logical biconditional. ...

where z is in K_n and for all 0 ≤ j < n, the dimension [K_{j + 1} : K_j] = 2.

Impossible constructions

The algebraic characterization of constructible numbers provides an important necessary condition for constructibility: if z is constructible, then it is algebraic, and its minimal irreducible polynomial has degree a power of 2, or equivalently, the field extension Q(z)/Q has dimension a power of 2. One should note that it is true, (but not obvious to show) that the converse is false — this is not a sufficient condition for constructibility. However, this defect can be remedied by considering the normal closure of Q(z)/Q.

The nonconstructibility of certain numbers proves the impossibility of certain problems attempted by the philosophers of ancient Greece. In the following chart, each row represents a specific ancient construction problem. The left column gives the name of the problem. The second column gives an equivalent algebraic formulation of the problem. In other words, the solution to the problem is affirmative if and only if each number in the given set of numbers is constructible. Finally, the last column provides the simplest known counterexample. In other words, the number in the last column is an element of the set in the same row, but is not constructible. Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Ancient Greece is the period of Greek history spanning much of the Mediterranean and Black Sea basins and lasting for close to a millennium, until the rise of Christianity. ... It has been suggested that this article or section be merged with Logical biconditional. ... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

Construction problem	Associated set of numbers	Counterexample
Doubling the cube		is not constructible, because its minimal polynomial has degree 3 over Q
Trisecting the angle		is not constructible, because has minimal polynomial of degree 3 over Q
Squaring the circle		is not constructible, because is not algebraic over Q
Constructing all regular polygons		e2πi / 7 is not constructible, because 7 is not a Fermat prime

Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. ... A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... This square and circle have the same area. ... In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ...

See also

• Definable number