PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (231)
Orphanage
Unclass'd
Unproven (514)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] multiples of an algebraic number (Theorem)
Theorem 1   If $ \alpha$ is an algebraic number, then there exists a non-zero multiple of $ \alpha$ which is an algebraic integer.

Proof. Let $ \alpha$ be a root of the equation

$\displaystyle x^n\!+\!r_1x^{n-1}\!+\!r_2x^{n-2}\!+\cdots+\!r_n = 0,$
where $ r_1$, $ r_2$, ..., $ r_n$ are rational numbers ($ n > 0$). Let $ l$ be the least common multiple of the denominators of the $ r_j$'s. Then we have
$\displaystyle 0 = l^n(\alpha^n\!+\!r_1\alpha^{n-1}\!+\!r_2\alpha^{n-2}\!+\cdots... ...l\alpha)^n\!+\!lr_1(l\alpha)^{n-1}\!+\!l^2r_2(l\alpha)^{n-2}\!+\cdots+\!l^nr_n,$
i.e. the multiple $ l\alpha$ of $ \alpha$ satisfies the algebraic equation
$\displaystyle x^n\!+\!lr_1x^{n-1}\!+\!l^2r_2x^{n-2}\!+\cdots+\!l^nr_n = 0$
with rational integer coefficients.

According to the theorem, any algebraic number $ \xi$ is a quotient of an algebraic integer (of the field $ \mathbb{Q}(\xi)$) and a rational integer.



"multiples of an algebraic number" is owned by pahio.
(view preamble | get metadata)

View style:

See Also: theory of algebraic and transcendental numbers


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: field, quotient, coefficients, rational integer, algebraic equation, denominators, least common multiple, rational numbers, equation, proof, algebraic integer, algebraic number
There are 2 references to this entry.

This is version 5 of multiples of an algebraic number, born on 2005-05-30, modified 2008-05-10.
Object id is 7126, canonical name is MultiplesOfAnAlgebraicNumber.
Accessed 1280 times total.

Classification:
AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)