Field theory and polynomials Galois theory Theorem

Impossibility of the classical construction problems

Builds on Constructible number The degree of a constructible number is a power of two Impossibility of doubling the cube Impossibility of trisecting the angle Impossibility of squaring the circle Lindemann–Weierstrass theorem

Statement

Start from two points at distance $1$ in the plane, and use only straightedge and compass.

It is impossible to construct a segment of length $\sqrt[3]{2}$ , that is, to double the volume of a cube with edge $1$ .
It is impossible to trisect every angle; in particular the angle $\pi/3$ cannot be trisected.
It is impossible to construct a square with the same area as a circle of radius $1$ , that is, a segment of length $\sqrt{\pi}$ .

Remarks

A length is constructible exactly when it is a constructible number, and the degree of a constructible number over $\Q$ is a power of $2$ . For clause 1, $\sqrt[3]{2}$ has minimal polynomial $x^3 - 2$ , of degree $3$ ; see impossibility of doubling the cube. For clause 2, $2\cos(\pi/9)$ has minimal polynomial $x^3 - 3x - 1$ , of degree $3$ ; see impossibility of trisecting the angle. For clause 3, constructible numbers are algebraic, while $\pi$ , and hence $\sqrt{\pi}$ , is transcendental by the Lindemann–Weierstrass theorem; see impossibility of squaring the circle. Some angles can be trisected, such as $\pi/2$ , since $\pi/6$ is constructible; clause 2 says that no method works for all angles.

Going deeper

For a student

Each straightedge-and-compass step solves linear or quadratic equations, so each new point lies in an extension of degree $1$ or $2$ of the field of the previous points. A constructible number therefore lies at the top of a tower of quadratic extensions, and by the tower law its degree is a power of $2$ . Clauses 1 and 2 then follow from degree $3$ minimal polynomials, while clause 3 needs the much deeper transcendence of $\pi$ . The same method, with cyclotomic extensions, decides which regular polygons are constructible.

Reading path

To understand this page from the ground up, read these 82 pages in order. The foundations come first.

Algebraic number definition
Group concept
Straightedge and compass construction concept
Transcendental number definition
Ring definition
Constructible number definition
Doubling the cube problem
Squaring the circle problem
Trisecting the angle problem
Lindemann–Weierstrass theorem statement
Commutative ring definition
Ideal definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Divisibility in a commutative ring definition
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Degree of a polynomial definition
Polynomial evaluation definition
Field definition
Integral domain definition
Kernel of a ring homomorphism definition
Ring isomorphism definition
Principal ideal definition
Quotient ring definition
Associate elements definition
Prime element definition
Leading coefficient definition
The only ideals of a field are zero and the whole field statement
Evaluation homomorphism definition
Root of a polynomial definition
Subfield definition
Field of fractions definition
Irreducible element definition
Euclidean domain definition
Correspondence theorem for rings statement
Greatest common divisor in an integral domain definition
Principal ideal domain definition
Monic polynomial definition
Universal property of the quotient ring statement
Degree of a product statement
The constructible numbers form a field closed under square roots statement
Unique factorisation domain definition
Irreducible polynomial definition
Field extension definition
Rational root theorem statement
R/M is a field iff M is maximal statement
Division algorithm for polynomials statement
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Content of a polynomial definition
R[x] is an integral domain iff R is statement
First isomorphism theorem for rings statement
Algebraic element definition
Intermediate field definition
Degree of a field extension definition
Primitive polynomial definition
Remainder theorem statement
Minimal polynomial of an algebraic element definition
K[x] is a Euclidean domain statement
Transcendental element definition
Tower law statement
Field generated by adjoining elements definition
The minimal polynomial divides every polynomial vanishing at the element statement
K[x]/(f) is a field iff f is irreducible statement
Product of primitive polynomials is primitive statement
Factor theorem statement
Transcendence of pi statement
Simple extension definition
K(a) is isomorphic to K[x] modulo the minimal polynomial statement
Gauss's lemma statement
Polynomials of degree two or three are irreducible iff they have no roots statement
Eisenstein's criterion statement
Degree of a simple extension is the degree of the minimal polynomial statement
The degree of a constructible number is a power of two statement
Impossibility of doubling the cube statement
Impossibility of squaring the circle statement
Impossibility of trisecting the angle statement
Impossibility of the classical construction problems statement

Tag	`W8J42M1G28`
URI	`m2://W8J42M1G28`
Kind	statement, theorem
Writing	W0 stub
MSC	12F05, 51M15
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About	Constructible number, Straightedge and compass construction
Uses	Constructible number, The degree of a constructible number is a power of two, Impossibility of doubling the cube, Impossibility of trisecting the angle, Impossibility of squaring the circle, Lindemann–Weierstrass theorem
Changed	2 November 2026