Editing Constructible polygon (section)

==General theory==

In the light of later work on [[Galois theory]], the principles of these proofs have been clarified. It is straightforward to show from [[analytic geometry]] that constructible lengths must come from base lengths by the solution of some sequence of [[quadratic equation]]s.<ref>{{citation
 | last = Cox | first = David A. | authorlink = David A. Cox
 | contribution = Theorem 10.1.6
 | doi = 10.1002/9781118218457
 | edition = 2nd
 | isbn = 978-1-118-07205-9
 | page = 259
 | publisher = John Wiley & Sons
 | series = Pure and Applied Mathematics
 | title = Galois Theory
 | year = 2012}}.</ref> In terms of [[field theory (mathematics)|field theory]], such lengths must be contained in a [[field extension]] generated by a tower of [[quadratic extension]]s. It follows that a field generated by constructions will always have [[degree of a field extension|degree]] over the base field that is a power of two.

In the specific case of a regular ''n''-gon, the question reduces to the question of [[constructible number|constructing a length]]

:cos&thinsp;{{sfrac|2{{pi}}|''n''}} ,

which is a [[trigonometric number]] and hence an [[algebraic number]]. This number lies in the ''n''-th [[cyclotomic field]] &mdash; and in fact in its [[real number|real]] [[field extension|subfield]], which is a [[Totally real number field|totally real field]] and a [[rational number|rational]] [[vector space]] of [[Hamel dimension|dimension]]

:½&thinsp;φ(''n''),

where φ(''n'') is [[Euler's totient function]]. Wantzel's result comes down to a calculation showing that φ(''n'') is a power of 2 precisely in the cases specified.

As for the construction of Gauss, when the [[Galois group]] is a 2-group it follows that it has a sequence of [[subgroup]]s of orders

:1, 2, 4, 8, ...

that are nested, each in the next (a [[composition series]], in [[group theory]] terminology), something simple to prove by [[mathematical induction|induction]] in this case of an [[abelian group]]. Therefore, there are subfields nested inside the cyclotomic field, each of degree 2 over the one before. Generators for each such field can be written down by [[Gaussian period]] theory. For example, for [[heptadecagon|''n''&nbsp;=&thinsp;17]] there is a period that is a sum of eight [[roots of unity]], one that is a sum of four roots of unity, and one that is the sum of two, which is

:cos&thinsp;{{sfrac|2{{pi}}|17}} .

Each of those is a root of a [[quadratic equation]] in terms of the one before. Moreover, these equations have [[real number|real]] rather than [[complex number|complex]] roots, so in principle can be solved by geometric construction: this is because the work all goes on inside a totally real field.

In this way the result of Gauss can be understood in current terms; for actual calculation of the equations to be solved, the periods can be squared and compared with the 'lower' periods, in a quite feasible algorithm.