See also: Theorem of Gauss

In Geometry, the Théorème of Gauss-Wantzel specifies the Requirement and sufficient so that a regular Polygone is constructible with the rule and the compass.

Gauss had had a presentiment of this requirement and sufficient but in 1796 qu ' an implication had not shown: If a regular polygon has sides and if is a power of 2 or is the product of a power of 2 and numbers of different Fermat first then this polygon is constructible. It is an analysis on the cyclotomic polynomials which allows the demonstration of this implication. One finds the details in the associated article there. It had left the Réciproque in the form of a conjecture.

Pierre-Laurent Wantzel in his publication of 1837 shows the reciprocal one thanks to its requirement so that a number is Constructible (Théorème of Wantzel). It is the concept of quadratic Tour of extension which makes it possible to bring the proof of it.

Number 5 is of Fermat because it is first and is written 2 ² + 1. Thus the construction of a realizable polygon of 5 east coasts. Such a constructible polygon at 20 east coasts since it is enough to leave the polygon to 5 sides to take the bisectrix of each angle (with the rule and the compass) to obtain a polygon at 10 sides and to start again the operation. And a polygon on 15 sides also because 15 is the product of two numbers of Fermat. Euclide had already established a construction besides of it.

Number 17 is of Fermat because is written 2⁴ + 1 and it is first. Polygon with 17 east coasts such a constructible and Gauss gave a method of construction of it.

On the other hand, 7 is not a number of Fermat and the regular Heptagone is not constructible. 9 = 3 ² is the square of a number of Fermat the first thus regular ennéagone is not constructible (cf quadratic Tour of extension).

Detailed results

The five numbers of Fermat are:

F ₀ = 3, F ₁ = 5, F ₂ = 17, F ₃ = 257, and F ₄ = 65537

.

Thus a polygon with N constructible east coast if:

N = 3,4,5,6,8,10,12,15,16,17,20,24,…

,

While it is not constructible with the rule and the compass if:

N = 7,9,11,13,14,18,19,21,22,23,25,…

.

Application

• Division of a disc on the identical surfaces of forms and equal surfaces

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