constructible number

Let $O$ and $A$ be two given distinct points in the Euclidean plane, and define $S$ to be the set of points that can be constructed with compass and straightedge starting with $O$ and $A$. Then the points of $S$ are called constructible points. $O$ and $A$ are, by definition, elements of $S$. To more precisely describe the remaining elements of $S$, make the following two definitions:{{sfnp|Kazarinoff|2003|p=10}}

• a line segment whose endpoints are in $S$ is called a constructed segment, and
• a circle whose center is in $S$ and which passes through a point of $S$ (alternatively, whose radius is the distance between some pair of distinct points of $S$) is called a constructed circle.

Then, the points of $S$, besides $O$ and $A$ are:{{sfnmp|Kazarinoff|2003|1p=10|Martin|1998|2pp=30–31|2loc=Definition 2.1}}

• the intersection of two non-parallel constructed segments, or lines through constructed segments,
• the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or
• the intersection points of two distinct constructed circles.

As an example, the midpoint of constructed segment $OA$ is a constructible point. One construction for it is to construct two circles with $OA$ as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment $OA$ is the point where this segment is crossed by the constructed line.This construction for the midpoint is given in Book I, Proposition 10 of Euclid's Elements.

The starting information for the geometric formulation can be used to define a Cartesian coordinate system in which the point $O$ is associated to the origin having coordinates $(0,0)$ and in which the point $A$ is associated with the coordinates $(1,\; 0)$. The points of $S$ may now be used to link the geometry and algebra by defining a constructible number to be a coordinate of a constructible point.{{sfnp|Kazarinoff|2003|p=18}}

Equivalent definitions are that a constructible number is the $x$-coordinate of a constructible point $(x,0)${{sfnp|Martin|1998|pp=30–31|loc=Definition 2.1}} or the length of a constructible line segment.{{harvp|Herstein|1986|p=237}}. To use the length-based definition, it is necessary to include the number zero as a constructible number, as a special case. In one direction of this equivalence, if a constructible point has coordinates $(x,y)$, then the point $(x,0)$ can be constructed as its perpendicular projection onto the $x$-axis, and the segment from the origin to this point has length $x$. In the reverse direction, if $x$ is the length of a constructible line segment, then intersecting the $x$-axis with a circle centered at $O$ with radius $x$ gives the point $(x,0)$. It follows from this equivalence that every point whose Cartesian coordinates are geometrically constructible numbers is itself a geometrically constructible point. For, when $x$ and $y$ are geometrically constructible numbers, point $(x,y)$ can be constructed as the intersection of lines through $(x,0)$ and $(0,y)$, perpendicular to the coordinate axes.{{sfnmp|Moise|1974|1p=227|Martin|1998|2p=33|2loc=Theorem 2.4}}

The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers. Even more simply, at the expense of making these formulas longer, the integers in these formulas can be restricted to be only 0 and 1.{{sfnp|Martin|1998|pp=36–37}} For instance, the square root of 2 is constructible, because it can be described by the formulas $\backslash sqrt2$ or $\backslash sqrt\{1+1\}$.

Analogously, the algebraically constructible complex numbers are the subset of complex numbers that have formulas of the same type, using a more general version of the square root that is not restricted to positive numbers but can instead take arbitrary complex numbers as its argument, and produces the principal square root of its argument. Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers.{{sfnp|Roman|1995|p=207}} For instance, the complex number $i$ has the formulas $\backslash sqrt\{-1\}$ or $\backslash sqrt\{0-1\}$, and its real and imaginary parts are the constructible numbers 0 and 1 respectively.

These two definitions of the constructible complex numbers are equivalent.{{harvp|Lawrence|Zorzitto|2021|p= [https://books.google.com/books?id=-koyEAAAQBAJ&pg=PA440 440]}}. In one direction, if $q=x+iy$ is a complex number whose real part $x$ and imaginary part $y$ are both constructible real numbers, then replacing $x$ and $y$ by their formulas within the larger formula $x+y\backslash sqrt\{-1\}$ produces a formula for $q$ as a complex number. In the other direction, any formula for an algebraically constructible complex number can be transformed into formulas for its real and imaginary parts, by recursively expanding each operation in the formula into operations on the real and imaginary parts of its arguments, using the expansionsFor the addition and multiplication formula, see {{harvp|Kay|2021|p=187|loc=Theorem 8.1.10}}. For the division formula, see {{harvp|Kay|2021|pp= 188, 224|loc= Equations 8.8 & 9.2}}. The expansion of the square root can be derived from the half-angle formula of trigonometry; see an equivalent formula at {{harvp|Lawrence|Zorzitto|2021|p= [https://books.google.com/books?id=-koyEAAAQBAJ&pg=PA440 440]}}.

• $(a+ib)\backslash pm\; (c+id)=(a\; \backslash pm\; c)+i(b\; \backslash pm\; d)$
• $(a+ib)(c+id)=(ac-bd)\; +\; i(ad+bc)$
• $\backslash frac\{1\}\{a+ib\}=\backslash frac\{a\}\{a^2+b^2\}\; +\; i\; \backslash frac\{-b\}\{a^2+b^2\}$
• $\backslash sqrt\{a+ib\}\; =\; \backslash frac\{(a+r)\backslash sqrt\{r\}\}\{s\}\; +\; i\backslash frac\{b\backslash sqrt\{r\}\}\{s\}$, where $r=\backslash sqrt\{a^2+b^2\{\}\_\{\backslash !\}\}$ and $s=\backslash sqrt\{(a+r)^2+b^2\}$.

The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers. Alternatively, they may be defined as the points in the complex plane given by algebraically constructible complex numbers. By the equivalence between the two definitions for algebraically constructible complex numbers, these two definitions of algebraically constructible points are also equivalent.

If $a$ and $b$ are the non-zero lengths of geometrically constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths $a+b$, $|a-b|$, $ab$, and $a/b$. The latter two can be done with a construction based on the intercept theorem. A slightly less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length $\backslash sqrt\{a\}$ from a constructed segment of length $a$. It follows that every algebraically constructible number is geometrically constructible, by using these techniques to translate a formula for the number into a construction for the number.{{harvp|Herstein|1986|pp=236–237}}; {{harvp|Moise|1974|p=224}}; {{harvp|Fraleigh|1994|pp=426–427}}; {{harvp|Courant|Robbins|1996|pp=120–122|loc=Section III.1.1: Construction of fields and square root extraction}}.

{{multiple image|total_width=720|align=center|header=Compass and straightedge constructions for constructible numbers

|image1=Number construction multiplication.svg|caption1=$ab$ based on the intercept theorem

|image2=Number construction division.svg|caption2=$\backslash frac\{a\}\{b\}$ based on the intercept theorem

|image3=Root_construction_geometric_mean5.svg|caption3=$\backslash sqrt\{p\}$ based on the geometric mean theorem}}

In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and $y$-intercept for lines, and center and radius for circles. It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object that might be added in a single step of a compass-and-straightedge construction. It follows from these formulas that every geometrically constructible number is algebraically constructible.{{sfnmp|1a1=Martin|1y=1998|1pp=38–39|2a1=Courant|2a2=Robbins|2y=1996|2pp=131–132}}

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. More specifically, the constructible real numbers form a Euclidean field, an ordered field containing a square root of each of its positive elements.{{sfnp|Martin|1998|p=35|loc=Theorem 2.7}} Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible.

It is convenient to consider, in place of the whole field of constructible numbers, the subfield $\backslash mathbb\{Q\}(\backslash gamma)$ generated by any given constructible number $\backslash gamma$, and to use the algebraic construction of $\backslash gamma$ to decompose this field. If $\backslash gamma$ is a constructible real number, then the values occurring within a formula constructing it can be used to produce a finite sequence of real numbers $\backslash alpha\_1,\backslash dots,\; \backslash alpha\_n=\backslash gamma$ such that, for each $i$, $\backslash mathbb\{Q\}(\backslash alpha\_1,\backslash dots,\backslash alpha\_i)$ is an extension of $\backslash mathbb\{Q\}(\backslash alpha\_1,\backslash dots,\backslash alpha\_\{i-1\})$ of degree 2.{{sfnp|Fraleigh|1994|p=429}} Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite tower of real quadratic extensions,

$$\backslash mathbb\{Q\}\; =\; K\_0\; \backslash subseteq\; K\_1\; \backslash subseteq\; \backslash dots\; \backslash subseteq\; K\_n,$$

starting with the rational field $\backslash mathbb\{Q\}$ where $\backslash gamma$ is in $K\_n$ and for all , $[K\_j:K\_\{j-1\}]=2$.{{sfnp|Roman|1995|p=59}} It follows from this decomposition that the degree of the field extension $[\backslash mathbb\{Q\}(\backslash gamma):\backslash mathbb\{Q\}]$ is $2^r$, where $r$ counts the number of quadratic extension steps.{{sfnp|Neumann|1998}}

Analogously to the real case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions.{{sfnp|Rotman|2006|p=361}} More precisely, $\backslash gamma$ is constructible if and only if there exists a tower of fields

$$\backslash mathbb\{Q\}\; =\; F\_0\; \backslash subseteq\; F\_1\; \backslash subseteq\; \backslash dots\; \backslash subseteq\; F\_n,$$

where $\backslash gamma$ is in $F\_n$, and for all

To obtain a sufficient condition for constructibility, one must instead consider the splitting field $K=\backslash mathbb\{Q\}(\backslash gamma,\backslash gamma\text{'},\backslash gamma\text{'}\text{'},\backslash dots)$ obtained by adjoining all roots of the minimal polynomial of $\backslash gamma$. If the degree of {{em|this}} extension is a power of two, then its Galois group $G=\backslash mathrm\{Gal\}(K/\backslash mathbb\{Q\})$ is a 2-group, and thus admits a descending sequence of subgroups

$$G\; =\; G\_n\; \backslash supseteq\; G\_\{n-1\}\; \backslash supseteq\; \backslash cdots\; \backslash supseteq\; G\_0\; =\; 1,$$

with $|G\_k|\; =\; 2^k$ for $0\backslash leq\; k\; \backslash leq\; n.$ By the fundamental theorem of Galois theory, there is a corresponding tower of quadratic extensions

$$\backslash mathbb\{Q\}\; =\; F\_0\; \backslash subseteq\; F\_1\; \backslash subseteq\; \backslash dots\; \backslash subseteq\; F\_n\; =\; K,$$

whose topmost field contains $\backslash gamma,$ and from this it follows that $\backslash gamma$ is constructible.

The fields that can be generated from towers of quadratic extensions of $\backslash mathbb\{Q\}$ are called {{dfn|iterated quadratic extensions}} of $\backslash mathbb\{Q\}$. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of $\backslash mathbb\{Q\}$.{{sfnp|Martin|1998|p=37|loc=Theorem 2.10}}

{{main|Trigonometric number}}

Trigonometric numbers are the cosines or sines of angles that are rational multiples of $\backslash pi$. These numbers are always algebraic, but they may not be constructible. The cosine or sine of the angle $2\backslash pi/n$ is constructible only for certain special numbers $n$:{{sfnp|Martin|1998|p=46}}

• The powers of two
• The Fermat primes, prime numbers that are one plus a power of two
• The products of powers of two and any number of distinct Fermat primes.

Thus, for example, $\backslash cos(\backslash pi/15)$ is constructible because 15 is the product of the Fermat primes 3 and 5; but $\backslash cos(\backslash pi/9)$ is not constructible (not being the product of {{em|distinct}} Fermat primes) and neither is $\backslash cos(\backslash pi/7)$ (being a non-Fermat prime).

{{multiple image|total_width=600

|image1=01-Würfelverdoppelung-Menaichmos-1.svg|caption1=A cube and its double

|image2=01 Dreiteilung-Winkel Einleitungsbild.svg|caption2=An angle and its trisection

|image3=SquareCircle.svg|caption3=Circle and square with equal areas}}

The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable.{{sfnp|Stewart|1989|p=51}} However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform.{{sfnp|Klein|1897|p=3}} (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way. One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.)The description of these alternative solutions makes up much of the content of {{harvp|Knorr|1986}}.

In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems:

;Doubling the cube

:The problem of doubling the unit square is solved by the construction of another square on the diagonal of the first one, with side length $\backslash sqrt2$ and area $2$. Analogously, the problem of doubling the cube asks for the construction of the length $\backslash sqrt[3]\{2\}$ of the side of a cube with volume $2$. It is not constructible, because the minimal polynomial of this length, $x^3-2$, has degree 3 over $\backslash Q$.{{sfnmp|Klein|1897|1p=13|Fraleigh|1994|2pp=429–430}} As a cubic polynomial whose only real root is irrational, this polynomial must be irreducible, because if it had a quadratic real root then the quadratic conjugate would provide a second real root.{{harvp|Courant|Robbins|1996|pp=134–135|loc=Section III.3.1: Doubling the cube}}

;Angle trisection

:In this problem, from a given angle $\backslash theta$, one should construct an angle $\backslash theta/3$. Algebraically, angles can be represented by their trigonometric functions, such as their sines or cosines, which give the Cartesian coordinates of the endpoint of a line segment forming the given angle with the initial segment. Thus, an angle $\backslash theta$ is constructible when $x=\backslash cos\backslash theta$ is a constructible number, and the problem of trisecting the angle can be formulated as one of constructing $\backslash cos(\backslash tfrac\{1\}\{3\}\backslash arccos\; x)$. For example, the angle $\backslash theta=\backslash pi/3=60^\backslash circ$ of an equilateral triangle can be constructed by compass and straightedge, with $x=\backslash cos\backslash theta=\backslash tfrac12$. However, its trisection $\backslash theta/3=\backslash pi/9=20^\backslash circ$ cannot be constructed, because $\backslash cos\backslash pi/9$ has minimal polynomial $8x^3-6x-1$ of degree 3 over $\backslash Q$. Because this specific instance of the trisection problem cannot be solved by compass and straightedge, the general problem also cannot be solved.{{harvp|Fraleigh|1994|pp=429–430}}; {{harvp|Courant|Robbins|1996|pp=137–138|loc=Section III.3.3: Trisecting the angle}}.

;Squaring the circle

:A square with area $\backslash pi$, the same area as a unit circle, would have side length $\backslash sqrt\backslash pi$, a transcendental number. Therefore, this square and its side length are not constructible, because it is not algebraic over $\backslash Q$.{{sfnp|Fraleigh|1994|pp=429–430}}

;Regular polygons

:If a regular $n$-gon is constructed with its center at the origin, the angles between the segments from the center to consecutive vertices are $2\backslash pi/n$. The polygon can be constructed only when the cosine of this angle is a trigonometric number. Thus, for instance, a 15-gon is constructible, but the regular heptagon is not constructible, because 7 is prime but not a Fermat prime.{{sfnp|Fraleigh|1994|p=504}} For a more direct proof of its non-constructibility, represent the vertices of a regular heptagon as the complex roots of the polynomial $x^7-1$. Removing the factor $x-1$, dividing by $x^3$, and substituting $y=x+1/x$ gives the simpler polynomial $y^3+y^2-2y-1$, an irreducible cubic with three real roots, each two times the real part of a complex-number vertex. Its roots are not constructible, so the heptagon is also not constructible.{{harvp|Courant|Robbins|1996|pp=138–139|loc=Section III.3.4: The regular heptagon}}.

;Alhazen's problem

:If two points and a circular mirror are given, where on the circle does one of the given points see the reflected image of the other? Geometrically, the lines from each given point to the point of reflection meet the circle at equal angles and in equal-length chords. However, it is impossible to construct a point of reflection using a compass and straightedge. In particular, for a unit circle with the two points $(\backslash tfrac16,\backslash tfrac16)$ and $(-\backslash tfrac12,\backslash tfrac12)$ inside it, the solution has coordinates forming roots of an irreducible degree-four polynomial $x^4-2x^3+4x^2+2x-1$. Although its degree is a power of two, the splitting field of this polynomial has degree divisible by three, so it does not come from an iterated quadratic extension and Alhazen's problem has no compass and straightedge solution.{{harvp|Neumann|1998}}. {{harvp|Elkin|1965}} comes to the same conclusion using different points and a different polynomial.

The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: doubling the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry.Plutarch, Quaestiones convivales [https://web.archive.org/web/20080815001514/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80 VIII.ii], 718ef. However, this attribution is challenged,{{sfnp|Kazarinoff|2003|p=28}} due, in part, to the existence of another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) that says that all three found solutions but they were too abstract to be of practical value.{{sfnp|Knorr|1986|p=4}} Proclus, citing Eudemus of Rhodes, credited Oenopides ({{circa}} 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction.{{sfnp|Knorr|1986|pp=15–17}} The restriction to compass and straightedge is essential to the impossibility of the classic construction problems. Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. The Quadratrix of Hippias of Elis, the conics of Menaechmus, or the marked straightedge (neusis) construction of Archimedes have all been used, as has a more modern approach via paper folding.{{sfnp|Friedman|2018|pp=1–3}}

Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is often treated alongside them. The Greeks knew how to construct regular {{nowrap|$n$-gons}} with $n=2^h$ (for any integer $h\backslash ge\; 2$), 3, 5, or the product of any two or three of these numbers, but other regular {{nowrap|$n$-gons}} eluded them. In 1796 Carl Friedrich Gauss, then an eighteen-year-old student, announced in a newspaper that he had constructed a regular 17-gon with straightedge and compass.{{sfnp|Kazarinoff|2003|p=29}} Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the {{em|sufficient}} condition for the construction of a regular {{nowrap|$n$-gon.}} Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably Felix Klein,{{sfnp|Klein|1897|p=16}} attributed this part of the proof to him as well.{{sfnp|Kazarinoff|2003|p=30}} Alhazen's problem is also not one of the classic three problems, but despite being named after Ibn al-Haytham (Alhazen), a medieval Islamic mathematician, it already appears in Ptolemy's work on optics from the second century.{{sfnp|Neumann|1998}}

Pierre Wantzel proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve using only compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible:

a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Gauss are also necessary).{{sfnmp|Wantzel|1837|Martin|1998|2p=46}} An attempted proof of the impossibility of squaring the circle was given by James Gregory in {{lang|la|Vera Circuli et Hyperbolae Quadratura|italic=unset}} (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of {{pi}}. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that {{pi}} is a transcendental number.{{sfnp|Martin|1998|p=44}}{{harvp|Klein|1897|pp=68–77|loc=Chapter IV}}: The transcendence of the number {{pi}}. Alhazen's problem was not proved impossible to solve by compass and straightedge until the work of Jack Elkin.{{harvp|Elkin|1965}}; see also {{harvp|Neumann|1998}} for an independent solution with more of the history of the problem.

The study of constructible numbers, per se, was initiated by René Descartes in La Géométrie, an appendix to his book Discourse on the Method published in 1637. Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.{{sfnp|Boyer|2004|pp=83–88}}

• Computable number
• Definable real number

{{reflist|30em}}

{{refbegin|30em}}

• {{citation |last= Boyer |first= Carl B. |date= 2004 |title= History of Analytic Geometry |orig-date= 1956 |publisher= Dover |isbn= 978-0-486-43832-0 |author-link= Carl Benjamin Boyer }}
• {{citation |last1= Courant |first1= Richard |last2= Robbins |first2= Herbert |date= 1996 |contribution= Chapter III: Geometrical constructions, the algebra of number fields |edition= 2nd |isbn= 0-19-510519-2 |pages= 117–164 |publisher= Oxford University Press |title= What is Mathematics? An elementary approach to ideas and methods |author1-link= Richard Courant |author2-link= Herbert Robbins }}
• {{citation |last= Elkin |first= Jack M. |date= March 1965 |title= A deceptively easy problem |journal= The Mathematics Teacher |pages= 194–199 |volume= 58 |issue= 3 |doi= 10.5951/MT.58.3.0194 |jstor= 27968003 }}
• {{citation |last= Fraleigh |first= John B. |date= 1994 |title= A First Course in Abstract Algebra |edition= 5th |publisher= Addison Wesley |isbn= 978-0-201-53467-2 }}
• {{citation |last= Friedman |first= Michael |date= 2018 |title= A History of Folding in Mathematics: Mathematizing the Margins |title-link= A History of Folding in Mathematics |series= Science Networks. Historical Studies |publisher= Birkhäuser |volume= 59 |isbn= 978-3-319-72486-7 }}
• {{citation |last= Herstein |first= I. N. |date= 1986 |title= Abstract Algebra |publisher= Macmillan |isbn= 0-02-353820-1 |author-link= Israel Nathan Herstein }}
• {{citation |last= Kay |first= Anthony |date= 2021 |title= Number Systems: A Path into Rigorous Mathematics |publisher= Taylor & Francis |isbn= 978-0-367-18065-2}}
• {{citation |last= Kazarinoff |first= Nicholas D. |date= 2003 |title= Ruler and the Round: Classic Problems in Geometric Constructions |orig-date= 1970 |publisher= Dover |isbn= 0-486-42515-0 |author-link= Nicholas D. Kazarinoff }}
• {{citation |last= Klein |first= Felix |title= Famous Problems of Elementary Geometry |date= 1897 |publisher= Ginn & Co |translator1-last= Beman |translator1-first= Wooster Woodruff |translator2-last= Smith |translator2-first= David Eugene |url= https://archive.org/details/famousproblemse00kleigoog |author-link= Felix Klein }}
• {{citation |last= Knorr |first= Wilbur Richard |date= 1986 |title= The Ancient Tradition of Geometric Problems |title-link= The Ancient Tradition of Geometric Problems |series= Dover Books on Mathematics |author-link= Wilbur Knorr |publisher= Courier Dover Publications |isbn= 978-0-486-67532-9 }}
• {{citation |last1= Lawrence |first1= John W. |last2= Zorzitto |first2= Frank A. |date= 2021 |title= Abstract Algebra: A Comprehensive Introduction |series= Cambridge Mathematical Textbooks |publisher= Cambridge University Press |isbn= 978-1-108-86551-7}}
• {{citation |last= Martin |first= George E. |date= 1998 |title= Geometric Constructions |title-link= Geometric Constructions |publisher= Springer-Verlag, New York |series= Undergraduate Texts in Mathematics |isbn= 0-387-98276-0 }}
• {{citation |last= Moise |first= Edwin E. |date= 1974 |title= Elementary Geometry from an Advanced Standpoint |edition= 2nd |publisher= Addison Wesley |isbn= 0-201-04793-4 |author-link= Edwin E. Moise }}
• {{citation |last= Neumann |first= Peter M. |date= 1998 |title= Reflections on reflection in a spherical mirror |journal= American Mathematical Monthly |issue= 6 |pages= 523–528 |volume= 105 |jstor= 2589403 |doi= 10.2307/2589403 |author-link= Peter M. Neumann }}
• {{citation |last= Roman |first= Steven |date= 1995 |title= Field Theory |publisher= Springer-Verlag |isbn= 978-0-387-94408-1 |author-link= Steven Roman }}
• {{citation |last= Rotman |first= Joseph J. |date= 2006 |title= A First Course in Abstract Algebra with Applications |edition= 3rd |publisher= Prentice Hall |isbn= 978-0-13-186267-8 |author-link= Joseph J. Rotman }}
• {{citation |last= Stewart |first= Ian |date= 1989 |title= Galois Theory |edition= 2nd |publisher= Chapman and Hall |isbn= 978-0-412-34550-0 |author-link= Ian Stewart (mathematician) }}
• {{citation |last= Wantzel |first= P. L. |date= 1837 |title= Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas |journal= Journal de Mathématiques Pures et Appliquées |language= fr |volume= 1 |issue= 2 |pages= 366–372 |url= http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16381&Deb=374&Fin=380&E=PDF |author-link= Pierre Wantzel }}

{{refend}}

{{commons category|Constructible numbers}}

• {{MathWorld |title=Constructible Number |urlname=ConstructibleNumber|mode=cs2|ref=none}}
• [http://www.cut-the-knot.org/arithmetic/rational.shtml Constructible Numbers] at Cut-the-knot

{{Algebraic numbers}}

{{Number systems}}

Category:Euclidean plane geometry

Category:Algebraic numbers