Constructible Number

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.

All Rational Numbers are constructible, and all constructible numbers are Algebraic Numbers (Courant and Robbins 1996, p. 133). If a Cubic Equation with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins, p. 136).

In particular, let $F_0$ be the Field of Rational Numbers. Now construct an extension field $F_1$ of constructible numbers by the adjunction of $\sqrt{k_0}$, where $k_0$ is in $F_0$, but $\sqrt{k_0}$ is not, consisting of all numbers of the form $a_0+b_0\sqrt{k_0}$, where $a_0,b_0\in F_0$. Next, construct an extension field $F_2$ of $F_1$ by the adjunction of $\sqrt{k_1}$, defined as the numbers $a_1+b_1\sqrt{k_1}$, where $a_1,b_1\in F_1$, and $k_1$ is a number in $F_1$ for which $\sqrt{k_1}$ does not lie in $F_1$. Continue the process $n$ times. Then constructible numbers are precisely those which can be reached by such a sequence of extension fields $F_n$, where $n$ is a measure of the ``complexity'' of the construction (Courant and Robbins 1996).

See also Algebraic Number, Compass, Constructible Polygon, Euclidean Number, Rational Number, Straightedge

References

Courant, R. and Robbins, H. ``Constructible Numbers and Number Fields.'' §3.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 127-134, 1996.