theorem on constructible numbers
Theorem 1.
Let be the field of constructible numbers and . Then there exists a nonnegative integer such that .
Before proving this theorem, some preliminaries must be addressed.
First of all, within this entry, the following nonconventional definition will be used:
Let be a subset of that contains a nonzero complex number and . Then is immediately constructible from if any of the following hold:
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for some ;
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for some ;
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for some ;
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for some with ;
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for some with and with .
The following lemmas are clear from this definition:
Lemma 1.
Let be a subset of that contains a nonzero complex number and . Then is constructible from if and only if there exists a finite sequence such that is immediately constructible from , is immediately constructible from , , and is immediately constructible from .
Lemma 2.
Let be a subfield of and . If is immediately constructible from , then either or .
Now to prove the theorem.
Proof.
By the first lemma, there exists a finite sequence such that is immediately constructible from , is immediately constructible from , , and is immediately constructible from . Thus, is immediately constructible from , , and is immediately constructible from . By the second lemma, is equal to either or , is equal to either or , , and is equal to either or . Therefore, there exists a nonnegative integer such that . Since , it follows that there exists a nonnegative integer such that . ∎
Title | theorem on constructible numbers |
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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 |