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A real number r ? R is called constructible if there is a finite sequence of compass-and-straightedge constructions that, when performed in order, will always create a point P with at least one coordinate equal to r. We showed above that 2 is constructible, and claim that n is constructible here: Theorem.

The set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number. This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers.

The Constructible Number Theorem: Every number ? that you can con- struct has the following properties: (i) ? is an algebraic number. (ii) The degree of the characteristic polynomial of ? is a power of 2.

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.

The constructible numbers are a subset of the algebraic numbers and this is a countable set because the algebric numbers are roots of a polynomial and the set of polynomials is countable and any polynomial has a finite number of roots.

Claim: If a and b are constructible numbers, then so is a + b. Proof: If either of a or b is zero, the claim is trivial. Otherwise, draw a line through the two points p, q that are distance |a| apart, then a circle of length |b| around q.

The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra. Thus, the constructible numbers (defined in any of the above ways) form a field.

Proof (A) Connecting two previously constructed points with a line. (B) Drawing a circle centered at an previously constructed point with an previously constructed radius. (C) Finding the intersection point of two previously constructed lines.

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