In mathematics, certain subsets of some fields are called orders. The set of integers is an order in the rational numbers (the only one). In an algebraic number field , an order is a ring of algebraic integers whose field of fractions is , and the maximal order, often denoted , is the ring of all algebraic integers in . In a non-Archimedean local field , an order is a subring which is generated by finitely many elements of non-negative valuation. In that case, the maximal order, denoted , is the valuation ring formed by all elements of non-negative valuation.
Giving the same name to such seemingly different notions is motivated by the localâÂÂglobal principle that relates properties of a number field with properties of all its local fields.
The definition of an order is somewhat context-dependent. The simplest definition is in an algebraic number field , where an order is a subring of that is a finitely-generated -module, which contains a rational basis of , i.e., such that
On the other hand, if is a non-archimedean local field, an order is a compact-open subring of . The maximal order in this case is the valuation ring of the field.
More generally, which includes both of these special cases, if an integral domain with fraction field , an -order in a finite-dimensional -algebra is a subring of which is a full -lattice; i.e. is a finite -module with the property that .
When ' is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings.
Some examples of orders are:
A fundamental property of -orders is that every element of an -order is integral over .
If the integral closure of in is an -order then the integrality of every element of every -order shows that must be the unique maximal -order in . However need not always be an -order: indeed need not even be a ring, and even if is a ring (for example, when is commutative) then need not be an -lattice.
The leading example is the case where ' is a number field ' and is its ring of integers. In algebraic number theory there are examples for any ' other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension ' of Gaussian rationals over , the integral closure of ' is the ring of Gaussian integers ' and so this is the unique maximal '-order: all other orders in ' are contained in it. For example, we can take the subring of complex numbers of the form , with and integers.
The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.