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 (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
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] canonical basis (Theorem)

Let $ \vartheta$ be an algebraic integer of degree $ n$. The algebraic number field $ \mathbb{Q}(\vartheta)$ has always an integral basis of the form

$ \displaystyle\omega_1 = 1,$
$ \displaystyle\omega_2 = \frac{a_{21}\!+\!\vartheta}{d_2},$
$ \displaystyle\omega_3 = \frac{a_{31}\!+\!a_{32}\vartheta\!+\!\vartheta^2}{d_3},$
$ \vdots\,\qquad\vdots\,\qquad\vdots$
$ \displaystyle\omega_n = \frac{a_{n1}\!+\!a_{n2}\vartheta\!+\ldots+\!a_{n,n-1}\vartheta^{n-2}\!+\!\vartheta^{n-1}}{d_n}$,

where the $ a_{ij}$'s and $ d_i$'s are rational integers such that

$\displaystyle d_2\mid d_3\mid d_4\mid\ldots\mid d_n,$
i.e.
$\displaystyle d_i\mid d_{i+1}\quad \forall\, i = 2,\,3,\,\ldots,\,n\!-\!1.$

The integral basis $ \omega_1,\,\omega_2,\,\ldots,\,\omega_n$ is called a canonical basis of the number field.

Remark. The integers $ a_{ij}$ can be reduced so that for all $ i$ and $ j$,

$\displaystyle -\frac{d_i}{2} < a_{ij} \leqq \frac{d_i}{2}.$
Then one may speak of an adjusted canonical basis. In the case of a quadratic number field $ \mathbb{Q}(\sqrt{d})$ with $ d \equiv 1\, ($mod$ \, 4)$ we have (see the examples of ring of integers of a number field)
$\displaystyle \omega_1 = 1, \quad \omega_2 = \frac{1\!+\!\sqrt{d}}{2}.$
The discriminant of this basis is $ d$.



"canonical basis" is owned by pahio.
(view preamble)

View style:

See Also: minimality of integral basis, examples of ring of integers of a number field, condition for power basis, integral basis of quadratic field

Also defines:  canonical basis, canonical basis of a number field, adjusted canonical basis

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

Cross-references: basis, discriminant, examples of ring of integers of a number field, quadratic number field, reduced, integers, rational, integral basis, algebraic number field, algebraic integer
There are 4 references to this entry.

This is version 10 of canonical basis, born on 2005-06-17, modified 2008-02-21.
Object id is 7165, canonical name is CanonicalBasis.
Accessed 2840 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 2 ]
Discussion
Style: Expand: Order:
forum policy
Canonical basis by pahio on 2005-06-17 18:47:14
If \theta^3 = 2, what is like a canonical basis of Q(\theta)?
[ reply | up ]
Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)