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## 6.4 Lattices and ideals

To generalise one step further, we can consider any finitely-generated subgroup M of K which contains a basis of K over ; such an M is called a lattice. Standard arguments then show that M is of the form . m1 + ... + . mn for some basis mi of K. For any fixed column vector v, let g be the invertible matrix that makes mi . v the standard basis of the space of column vectors. Then after applying g, we see that M . v becomes the standard lattice of column vectors with integer entries. The collection R(M) of all matrices in K that take M to itself, is thus identified with the ring which we denoted as Rg above. In the following paragraphs we assume that we have made this change of co-ordinates (i. e. that g is the identity matrix). In that case R = R(M) is precisely the order consisting of integer matrices. Moreover, there is a non-zero vector v so that M is precisely the collection of all so that . v is a vector with integer entries.

By collecting the denominators of the generators of M we can find a non-zero integer d so that d . M is contained in R. Since this is a subgroup of R that is closed under multiplication by R, it is an ideal I in R. Thus M = d-1I is a fractional ideal for R. It is clear that R(d . M) = R(M) = R. More generally, for any non-zero in K, we have R( . M) = R. Moreover, . M is obtained by replacing the v in the previous paragraph by v, which is just another non-zero vector.

Conversely, let I be a non-zero ideal in the ring R. Let be a non-zero element of I. Then is in K and by collecting the denominators we find a non-zero integer d so that d . has integer coefficients so is in R. But then d = d . is in I; thus I contains d . R. In particular, I contains a basis of K and is a free group of rank n; in other words I is a lattice. Clearly R is contained in R(I) but in general the latter could be bigger.

Now, for any non-zero ideal in R we have the restriction I = a. By the above discussion this is a non-zero ideal in . We also see that R/I is a quotient of the finite group R/aR; the latter group has order an. The order of R/I is called the norm of the ideal and denoted as Nm(I). The norm of an element is det(); these two definitions are related since Nm( . R) = | det()| (Exercise).

Now, we noted above that Nm(d . R) = dn for any positive integer d so we can extend the above definition by defining for M = d-1I, Nm(M) = d-nNm(I). Similarly, the restriction of d . R is clearly d, so we define the restriction of M to be d-1(I ). When M is contained in (i. e. M is an ideal) R, the two definitions are consistent.

Next: 6.5 Groups of invertible Up: 6 Algebraic Number Fields Previous: 6.3 Orders and Maximal
Kapil Hari Paranjape 2002-10-20