There is a natural symmetric pairing on n×n matrices given by
Now, R is a subgroup of the finitely-generated free abelian group of n×n matrices with integer coefficients; thus R is a finitely-generated free abelian group as well. If is any element of K we can clear denominators to find an integer d so that d is a matrix with integer entries. It follows that R contains a basis of K as a vector space over . Thus R is of the form . w1 + ... + . wn; moreover, K = . w1 + ... + . wn. Let denote the collection of all elements in K so that < , > is an integer for all in R. Finding such an is clearly equivalent to solving the system of equations
|r1 . w1, w1||+||...||+||rn . wn, w1||=||p1|
|r1 . w1, wn||+||...||+||rn . wn, wn||=||pn|
Now suppose R S, where S is another order (i. e. an Rg for some g). We clearly have the sequence of inclusions R S . It follows that DS divides DR; by decreasing induction we see that there is a maximal order. We also note that by duality, S/R and / have the same order, so that DR is the multiple of DS by the square of an integer. Let K be the collection of all elements of K whose characteristic polynomials have integer coefficients; one can show that this is closed under addition and multiplication. It is clear that K contains R since very matrix with integer entries has a characteristic polynomial with integer coefficients. By the above, we see that K is contained in , hence it is finitely generated; let k = . u1 + ... + . un. Let v be any non-zero column vector and consider the basis ui . v of the space of column vectors. With this change of basis, each each element of K is represented by a matrix with integer entries. Thus K is an order and the unique maximal order.
An extension of the example we looked at for fields is to associate an order with an irreducible polynomial P(T) = Tn + a1Tn - 1 + ... + an where the ai are all integers. We continue the notation of the previous subsection. It follows that is a matrix with integer coefficients; with a little effort one can also show that the natural order RP in () is precisely the collection of all integer linear combinations of the powers 1, , ..., . The discriminant of this order is also the discriminant of the polynomial P(T) and is denoted as DP. Unlike the case of fields, however, it is not true that every order has the form RP for some polynomial P(T).