| Maximal Order |
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#''O'' spans ''R'' over Q, so that Q''O'' = ''R''; and #''O'' is a Lattice in ''R''. The second condition can be stated more accurately, in terms of the extension of scalars of ''R'' to the Real Number s, embedding ''R'' in a real vector space (equivalently, taking the Tensor Product over ''Q''). In less formal terms, additively ''O'' should be a Free Abelian Group generated by a basis for ''R'' over Q. The leading example is the case where ''R'' is a Number Field ''K'' and ''O'' is its Ring Of Integers . In Algebraic Number Theory there are examples for any ''K'' other than the rational field of proper subrings of the ring of integers that are also orders. For example in the Gaussian Integer s we can take the subring of the a for which ''b'' is an Even Number . A basic result on orders states that the ring of integers in ''K'' is the unique maximal order: all other orders in ''K'' are contained in it. When ''R'' is not a s. Because there is a Local-global Principle For Lattices the maximal order question can be examined at a Local Field level. This technique is applied in algebraic number theory and Modular Representation Theory . |
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