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A point of the Euclidean plane is constructible if it can be obtained from $(0,0)$ and $(1,0)$ by finitely many straightedge-and-compass operations: draw a line through two obtained distinct points, draw a circle with obtained centre and radius equal to an obtained segment length, and retain isolated intersection points. Its two coordinates are constructible numbers. Conversely, constructible real coordinates can be placed on perpendicular axes and their coordinate lines intersected, giving the point. Coordinates created at each isolated intersection require only solving linear or quadratic equations over the current coordinate field; hence they lie in a real field extension obtained by a finite tower of quadratic extensions. Coincident curves supply no isolated new intersection points.

Consider the constructible angle whose cosine is $5/6$ and sine is $\sqrt{11}/6$. If one fifth of this angle were constructible, its cosine $c$ would satisfy the displayed polynomial, by the identity $\cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$. The identity follows by taking the real part of $(\cos \theta +i\sin \theta {)}^5$ and substituting ${\sin }^2\theta =1-{\cos }^2\theta$.

The polynomial is irreducible by Eisenstein's criterion at five, so $c$ has degree five over the rational numbers. The initial angle field $\mathbb{Q}(\sqrt{11})$ already has degree two, and adjoining any finite collection of constructible numbers gives total degree over the rational numbers a power of two. Such a field cannot contain the degree-five element $c$, by the tower law for field extensions. Hence this constructible angle cannot be divided into five equal angles by straightedge and compass construction. Some particular angles can be quintisected; the obstruction disproves a construction valid for all angles.

Doubling the volume of a unit cube requires constructing its new side length $\sqrt[3]{2}$. Its minimal polynomial $X^3-2$ is irreducible by Eisenstein's criterion at two, so this length has degree three. Every constructible number belongs to a finite tower of quadratic extensions, whose total degree is a power of two. The tower law for field extensions would force the degree three to divide that power of two, a contradiction. Thus no such straightedge and compass construction exists.

Begin with two points at distance one. A ruler draws the line through two known points; a compass draws a circle with a known centre and a constructible radius. New points are intersections of such lines and circles. A real number is constructible when it occurs as a coordinate of a point obtained by finitely many such operations; a complex point is constructible when both coordinates are.

If current coordinates lie in a field $K\subseteq \mathbb{R}$, a line-line intersection solves linear equations over $K$, while line-circle or circle-circle intersections reduce to a quadratic equation over $K$. Thus each step requires at most adjoining a square root. Consequently every constructible number lies in a towerConversely field sums and differences are obtained by laying off segments, products and quotients by similar triangles, and a positive square root by the geometric-mean construction in a semicircle. Hence every real number in such a tower is constructible. This proves the quadratic-tower criterion, not merely its necessity.

In particular a constructible algebraic number has degree a power of two. Degree alone is not sufficient in general; the exact criterion is containment in a quadratic tower. Equivalently the Galois group of its normal closure is a finite two-group: a tower's normal closure remains a two-extension, and a two-group admits a chain of subgroups with successive index two, yielding the converse by Galois correspondence.

The standard impossibility results follow. Doubling a unit-volume cube requires $\sqrt[3]{2}$, whose polynomial $X^3-2$ is irreducible by Eisenstein and has degree three. Trisecting an arbitrary angle would in particular trisect $60$ degrees, requiring $2\cos 20^{\circ }$, a root of $X^3-3X-1$. This cubic has no rational root and is irreducible, so this construction is impossible, although some particular angles can be trisected. Squaring the unit circle requires side length $\sqrt{\pi }$, impossible because $\pi$ is transcendental whereas constructible numbers are algebraic.

For a regular $n$-gon, constructibility is equivalent to that of $e^{2\pi i/n}$. The cyclotomic extension is Galois of degree $\phi (n)$, so it is constructible exactly when this degree is a power of two. Writing the prime factorization of $n$ shows the equivalent conditionIndeed odd prime exponents must be one, and each $p_i-1$ must be a power of two. If $2^m+1$ is prime, $m$ must itself be a power of two by factoring when $m$ has an odd divisor. Conversely this factorization makes the cyclotomic Galois group a two-group and provides a quadratic tower. This gives both positive constructions, such as the regular seventeen-gon, and obstructions, such as the regular seven-gon.

Identify $\mathbb{C}$ with the Euclidean plane and start with the points $0,1$. A complex number is a constructible number if a finite sequence of drawing lines through known points and circles with known centres and known distances as radii, and selecting their intersections, produces that point.

Line-line intersections solve linear equations in the current coordinate field. Line-circle and circle-circle intersections require at most one quadratic equation after eliminating a coordinate. Hence all coordinates obtained lie in a finite tower of real quadratic extensions of $\mathbb{Q}$. If $\alpha =x+iy$ is constructible, choose a common such real field $F$ containing $x,y$. Adjoining $i$ if needed still gives an extension $F(i)/\mathbb{Q}$ of power-of-two degree. By the tower law for field extensions,A regular heptagon centred at $0$ with one vertex $1$ would have $\xi =e^{2\pi i/7}$ as another constructible vertex. But its cyclotomic degree is $\phi (7)=6$, which is not a power of two. Thus the regular heptagon is not constructible.

The tower law for field extensions states that, for $F\subseteq K\subseteq L$ with both successive degrees finite,Indeed, if $u_i$ form an $F$-basis of $K$ and $v_j$ form a $K$-basis of $L$, their products $u_i{v}_j$ form an $F$-basis of $L$: expansion gives spanning, and independence follows first over $K$ and then over $F$.

Starting from a unit segment, every straightedge and compass construction introduces coordinates by solving linear or quadratic equations over the coordinates already available. Thus a constructible number belongs to a tower of field extensions whose successive degrees are one or two; the total degree is a power of two. Doubling the volume of a unit cube would require the edge length $\sqrt[3]{2}$. Its minimal polynomial over $\mathbb{Q}$ is $x^3-2$, irreducible by Eisenstein's criterion at the prime number two. Thus $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$. If that number lay in a construction field of degree $2^r$, the tower law would make three divide $2^r$, a contradiction. Therefore doubling the cube by straightedge and compass is impossible.

The tower law for field extensions states that, for $F\subseteq K\subseteq L$ with both successive degrees finite,Indeed, if $u_i$ form an $F$-basis of $K$ and $v_j$ form a $K$-basis of $L$, their products $u_i{v}_j$ form an $F$-basis of $L$: expansion gives spanning, and independence follows first over $K$ and then over $F$.

Starting from a unit segment, every straightedge and compass construction introduces coordinates by solving linear or quadratic equations over the coordinates already available. Thus a constructible number belongs to a tower of field extensions whose successive degrees are one or two; the total degree is a power of two. Doubling the volume of a unit cube would require the edge length $\sqrt[3]{2}$. Its minimal polynomial over $\mathbb{Q}$ is $x^3-2$, irreducible by Eisenstein's criterion at the prime number two. Thus $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$. If that number lay in a construction field of degree $2^r$, the tower law would make three divide $2^r$, a contradiction. Therefore doubling the cube by straightedge and compass is impossible.