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[parent] orders in a number field (Topic)

If $ \mu_1,\,\ldots,\,\mu_m$ are elements of an algebraic number field $ K$, then the subset

$\displaystyle M = \{n_1\mu_1+\ldots+n_m\mu_m\in K\,\vdots\;\; n_i\in\mathbb{Z}\;\;\forall i\}$
of $ K$ is a $ \mathbb{Z}$-module, called a module in $ K$. If the module contains as many over $ \mathbb{Z}$ linearly independent elements as is the degree of $ K$ over $ \mathbb{Q}$, then the module is complete.

If a complete module in $ K$ contains the unity 1 of $ K$ and is a ring, it is called an order (in German: Ordnung) in the field $ K$.

A number $ \alpha$ of the algebraic number field $ K$ is called a coefficient of the module $ M$, if $ \alpha M \subseteq M$.

Theorem 1. The set $ \mathcal{L}_M$ of all coefficients of a complete module $ M$ is an order in the field. Conversely, every order $ \mathcal{L}$ in the number field $ K$ is a coefficient ring of some module.

Theorem 2. If $ \alpha$ belongs to an order in the field, then the coefficients of the characteristic equation of $ \alpha$ and thus the coefficients of the minimal polynomial of $ \alpha$ are rational integers.

Theorem 2 means that any order is contained in the ring of integers of the algebraic number field $ K$. Thus this ring $ \mathcal{O}_K$, being itself an order, is the greatest order; $ \mathcal{O}_K$ is called the maximal order or the principal order (in German: Hauptordnung). The set of the orders is partially ordered by the set inclusion.

Example. In the field $ \mathbb{Q}(\sqrt{2})$, the coefficient ring of the module $ M$ generated by $ 2$ and $ \frac{\sqrt{2}}{2}$ is the module $ \mathcal{L}_M$ generated by $ 1$ and $ 2\sqrt{2}$. The maximal order of the field is generated by $ 1$ and $ \sqrt{2}$.

Bibliography

1
S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).



"orders in a number field" is owned by pahio.
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See Also: module

Also defines:  module, complete, order of a number field, principal order, maximal order
Keywords:  order

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basis of ideal in algebraic number field (Theorem) by pahio
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Cross-references: generated by, set inclusion, ring of integers, contained, integers, rational, minimal polynomial, coefficient, field, order, ring, unity, linearly independent, contains, subset, algebraic number field
There are 93 references to this entry.

This is version 14 of orders in a number field, born on 2007-03-30, modified 2008-02-24.
Object id is 9132, canonical name is OrdersInANumberField.
Accessed 2486 times total.

Classification:
AMS MSC: 06B10 (Order, lattices, ordered algebraic structures :: Lattices :: Ideals, congruence relations)
 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)
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No HTML for number order in fields by PrimeFan on 2007-03-30 17:08:12
TeX source shows, page images shows, but HTML with images is not showing on my computer. I'm using IE 6.0 on a Dell with Windows XP.
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