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 local–global principle that relates properties of a number field with properties of all its local fields.

Definitions

is a subring-module, which contains a rational basis

is a non-archimedean local field. The maximal order in this case is the valuation ring of the field.

an integral domain.

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.

Examples

Some examples of orders are:

• is the matrix ring

• a finite separable extension, then the integral closure.

• is an integral element, then the polynomial ring

• is the group ring of a finite group

-order is integral.

-lattice.

Algebraic number theory

The leading example is the case where is a number field 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, 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 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.

See also

• Hurwitz quaternion order – An example of ring order

Notes

References