theorem on constructible numbers


Theorem 1.

Let F be the field of constructible numbers and αF. Then there exists a nonnegative integer k such that [Q(α):Q]=2k.

Before proving this theorem, some preliminaries must be addressed.

First of all, within this entry, the following nonconventional definition will be used:

Let S be a subset of that contains a nonzero complex numberMathworldPlanetmathPlanetmath and α. Then α is immediately constructible from S if any of the following hold:

  • α=a+b for some a,bS;

  • α=a-b for some a,bS;

  • α=ab for some a,bS;

  • α=a/b for some a,bS with b0;

  • α=|z|eiθ2 for some zS with z0 and θ=arg(z) with 0θ<2π.

The following lemmas are clear from this definition:

Lemma 1.

Let S be a subset of C that contains a nonzero complex number and αC. Then α is constructible from S if and only if there exists a finite sequencePlanetmathPlanetmath α1,,αnC such that α1 is immediately constructible from S, α2 is immediately constructible from S{α1}, , and α is immediately constructible from S{α1,,αn}.

Lemma 2.

Let F be a subfieldMathworldPlanetmath of C and αC. If α is immediately constructible from F, then either [F(α):F]=1 or [F(α):F]=2.

Now to prove the theorem.

Proof.

By the first lemma, there exists a finite sequence α1,,αn such that α1 is immediately constructible from , α2 is immediately constructible from {α1}, , and α is immediately constructible from {α1,,αn}. Thus, α2 is immediately constructible from (α1), , and α is immediately constructible from (α1,,αn). By the second lemma, [(α1):] is equal to either 1 or 2, [(α1,α2):(α1)] is equal to either 1 or 2, , and [(α1,,αn,α):(α1,,αn)] is equal to either 1 or 2. Therefore, there exists a nonnegative integer m such that [(α1,,αn,α):]=2m. Since (α)(α1,,αn,α), it follows that there exists a nonnegative integer k such that [(α):]=2k. ∎

Title theorem on constructible numbers
Canonical name TheoremOnConstructibleNumbers
Date of creation 2013-03-22 17:16:28
Last modified on 2013-03-22 17:16:28
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 12D15
Related topic ConstructibleNumbers
Related topic ClassicalProblemsOfConstructibility
Defines immediately constructible from