PlanetMath: theorem on constructible numbers

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

theorem on constructible numbers

(Theorem)

Theorem Let $% latex2html id marker 395 $ \mathbb{F}$$ be the field of constructible numbers and $% latex2html id marker 397 $ \alpha\in\mathbb{F}$$ . Then there exists a nonnegative integer such that $% latex2html id marker 401 $ [\mathbb{Q}(\alpha)\!:\!\mathbb{Q}]=2^k$$ .

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 $% latex2html id marker 405 $ \mathbb{C}$$ that contains a nonzero complex number and $% latex2html id marker 407 $ \alpha\in\mathbb{C}$$ . Then $\alpha$ is immediately constructible from if any of the following hold:

$\alpha=a+b$ for some $a,b\in S$ ;
$\alpha=a-b$ for some $a,b\in S$ ;
$\alpha=ab$ for some $a,b\in S$ ;
$\alpha=a/b$ for some $a,b\in S$ with $b\neq 0$ ;
$% latex2html id marker 431 $ \alpha=\sqrt{\vert z\vert}e^{\frac{i\theta}{2}}$$ for some $z\in S$ with $z \neq 0$ and $% latex2html id marker 437 $ \theta=\operatorname{arg}(z)$$ with $% latex2html id marker 439 $ 0\le\theta <2\pi$$ .

The following lemmas are clear from this definition:

Lemma 1 Let be a subset of $% latex2html id marker 448 $ \mathbb{C}$$ that contains a nonzero complex number and $% latex2html id marker 450 $ \alpha\in\mathbb{C}$$ . Then $\alpha$ is constructible from if and only if there exists a finite sequence $% latex2html id marker 456 $ \alpha_1,\dots ,\alpha_n\in\mathbb{C}$$ such that $\alpha_1$ is immediately constructible from , $\alpha_2$ is immediately constructible from $S \cup \{\alpha_1\}$ , $\dots$ , and $\alpha$ is immediately constructible from $S \cup \{\alpha_1,\dots ,\alpha_n\}$ .

Lemma 2 Let be a subfield of $% latex2html id marker 479 $ \mathbb{C}$$ and $% latex2html id marker 481 $ \alpha\in\mathbb{C}$$ . If $\alpha$ is immediately constructible from , then either $[F(\alpha)\!:\!F]=1$ or $[F(\alpha)\!:\!F]=2$ .

Now to prove the theorem.

Proof. By the first lemma, there exists a finite sequence $% latex2html id marker 494 $ \alpha_1,\dots ,\alpha_n\in\mathbb{C}$$ such that $\alpha_1$ is immediately constructible from $% latex2html id marker 498 $ \mathbb{Q}$$ , $\alpha_2$ is immediately constructible from $% latex2html id marker 502 $ \mathbb{Q} \cup \{\alpha_1\}$$ , $\dots$ , and $\alpha$ is immediately constructible from $% latex2html id marker 508 $ \mathbb{Q} \cup \{\alpha_1,\dots ,\alpha_n\}$$ . Thus, $\alpha_2$ is immediately constructible from $% latex2html id marker 512 $ \mathbb{Q}(\alpha_1)$$ , $\dots$ , and $\alpha$ is immediately constructible from $% latex2html id marker 518 $ \mathbb{Q}(\alpha_1,\dots ,\alpha_n)$$ . By the second lemma, $% latex2html id marker 520 $ [\mathbb{Q}(\alpha_1)\!:\!\mathbb{Q}]$$ is equal to either

, $% latex2html id marker 526 $ [\mathbb{Q}(\alpha_1,\alpha_2)\!:\!\mathbb{Q}(\alpha_1)]$$ is equal to either

, $\dots$ , and $% latex2html id marker 534 $ [\mathbb{Q}(\alpha_1, \dots ,\alpha_n,\alpha)\!:\!\mathbb{Q}(\alpha_1, \dots ,\alpha_n)]$$ is equal to either

. Therefore, there exists a nonnegative integer

such that $% latex2html id marker 542 $ [\mathbb{Q}(\alpha_1, \dots ,\alpha_n,\alpha)\!:\!\mathbb{Q}]=2^m$$ . Since $% latex2html id marker 544 $ \mathbb{Q} \subseteq \mathbb{Q}(\alpha) \subseteq \mathbb{Q}(\alpha_1, \dots ,\alpha_n,\alpha)$$ , it follows that there exists a nonnegative integer

such that $% latex2html id marker 548 $ [\mathbb{Q}(\alpha)\!:\!\mathbb{Q}]=2^k$$ . $\qedsymbol$

"theorem on constructible numbers" is owned by Wkbj79.

(view preamble)

Also defines:

immediately constructible from

Log in to rate this entry.
(view current ratings)

Cross-references: subfield, finite sequence, constructible from, clear, complex number, contains, subset, integer, field of constructible numbers
There are 3 references to this entry.

This is version 7 of theorem on constructible numbers, born on 2007-06-18, modified 2007-06-23.
Object id is 9614, canonical name is TheoremOnConstructibleNumbers.
Accessed 693 times total.

Classification:

AMS MSC:

12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact