Ring of integers in a number field: Difference between revisions
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{{redirect|ring of integers|the ring of rational integers|ring of rational integers}} | |||
==Definition== | ==Definition== | ||
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The ring of integers in a number field is also termed a '''maximal order''', here an [[order in a number field|order]] is a subring of <math>K</math> which is free as a <math>\mathbb{Z}</math>-module, and which, over <math>\mathbb{Q}</math>, generates <math>K</math>. | The ring of integers in a number field is also termed a '''maximal order''', here an [[order in a number field|order]] is a subring of <math>K</math> which is free as a <math>\mathbb{Z}</math>-module, and which, over <math>\mathbb{Q}</math>, generates <math>K</math>. | ||
==Relation with other properties== | |||
===Weaker properties=== | |||
* [[Dedekind domain]] | |||
Latest revision as of 16:34, 12 May 2008
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
"ring of integers" redirects here. For the ring of rational integers, see ring of rational integers
Definition
Let be a number field. Then the ring of integers in , denoted , is the subring of comprising all those elements which are integral over , in other words, which satisfy monic polynomials with integer coefficients.
The ring of integers in a number field is also termed a maximal order, here an order is a subring of which is free as a -module, and which, over , generates .
