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[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.
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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

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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 2905 times total.

Classification:
AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
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Canonical basis by pahio on 2005-06-17 18:47:14
If \theta^3 = 2, what is like a canonical basis of Q(\theta)?
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