Field theory and polynomials Galois theory Theorem

Gauss–Wantzel theorem

Builds on Constructible number Irreducibility of the cyclotomic polynomials over the rationals The degree of a constructible number is a power of two

Statement

Let $n \ge 3$ . The regular $n$ -gon can be constructed with straightedge and compass if and only if $n = 2^k p_1 p_2 \cdots p_t$ , with $k, t \ge 0$ and $p_1, \dots, p_t$ distinct Fermat primes.

Remarks

A Fermat prime is a prime number of the form $2^{2^m} + 1$ . The known Fermat primes are $3, 5, 17, 257$ and $65537$ . The regular $n$ -gon is constructible if and only if $e^{2\pi i/n}$ is a constructible number. Its degree over $\mathbb{Q}$ is $\varphi(n)$ , by the irreducibility of $\Phi_n$ . The condition on $n$ holds if and only if $\varphi(n)$ is a power of $2$ . So the $17$ -gon and the $15$ -gon are constructible, but the $7$ -gon and the $9$ -gon are not.

History

Carl Friedrich Gauss proved in 1796 that the regular $17$ -gon is constructible. In his 1801 book Disquisitiones Arithmeticae, he gave the sufficient condition. Pierre Wantzel gave a full proof of the necessity in 1837.

Reading path

To understand this page from the ground up, read these 59 pages in order. The foundations come first.

Group concept
Straightedge and compass construction concept
Ring definition
Constructible number definition
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
The only ideals of a field are zero and the whole field statement
Evaluation homomorphism definition
Leading coefficient definition
Root of a polynomial definition
Subfield definition
Irreducible element definition
Euclidean domain definition
Correspondence theorem for rings statement
Principal ideal domain definition
The constructible numbers form a field closed under square roots statement
Monic polynomial definition
Universal property of the quotient ring statement
Irreducible polynomial definition
Degree of a product statement
Field extension definition
R/M is a field iff M is maximal statement
Cyclotomic polynomial definition
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Division algorithm for polynomials statement
Algebraic element definition
R[x] is an integral domain iff R is statement
First isomorphism theorem for rings statement
Intermediate field definition
Degree of a field extension definition
Irreducibility of the cyclotomic polynomials over the rationals statement
Minimal polynomial of an algebraic element definition
K[x] is a Euclidean domain statement
Field generated by adjoining elements definition
Tower law statement
The minimal polynomial divides every polynomial vanishing at the element statement
K[x]/(f) is a field iff f is irreducible statement
Simple extension definition
K(a) is isomorphic to K[x] modulo the minimal polynomial 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
Gauss–Wantzel theorem statement

Tag	`0Y34JMJNE9`
URI	`m2://0Y34JMJNE9`
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Uses	Constructible number, Irreducibility of the cyclotomic polynomials over the rationals, The degree of a constructible number is a power of two

Gauss–Wantzel theorem

Statement

Remarks

History

Reading path

Further reading

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