We now want to give a ``measure'' associated with an order *R*. The
space of *n*×*n* matrices with rational entries is naturally
contained in the space of *n*×*n* matrices with real entries. Thus
we can consider the ring
^{ . }*K* of real linear combinations of
elements of *K*. This is an *n*-dimensional vector space over
.
Thus, for any lattice *M*, the space
*T*_{M} = ^{ . }*K*/*M* is an
*n*-dimensional torus. Taking some translation invariant measure on
^{ . }*K* gives us a notion of volume for the tori *T*_{M} with the
property that
vol(*T*_{M}) = vol(*T*_{R})Nm(*M*). Now, if *A* is any (compact
measurable) subset of
^{ . }*K* with the property that
vol(*A*) > vol(*T*_{M}) then the map *A**T*_{M} cannot be one-to-one (with
a little thought it is clear that this is actually also true if
vol(*A*) vol(*T*_{M})). The difference between two points with the
same inverse image will give a non-zero element of *M*.

Now, one natural way to identify
^{ . }*K* with
^{n} (and thus
put a measure on it) is to use ``simultaneous diagonalisation''. As
seen above *K* is generated by a single *n*×*n* matrix
whose characteristic polynomial *P*(*T*) is irreducible over rationals.
This means that this has distinct roots and so over real numbers can
be brought into a ``diagonal'' form as below by a suitable change of
co-ordinates.

= *w*_{i}^{(1)},..., *w*_{i}^{(r1)}, Re, Im,..., Re, Im

Let the matrix
(complex entries) be given by
= *w*_{i}^{(1)},..., *w*_{i}^{(r1)},,,...,,

Standard rules for column operations on determinants show that the
determinant of
is 2
Now consider the region *A* consisting of all *x* in
^{ . }*K* so
that
| *x*^{(i)}| *a*_{i} and
|| *b*_{i} for some
positive constants *a*_{i} and *b*_{j}. We have

vol(*A*) = 2^{r1}*a*_{i}*b*_{j}^{2}

Thus, in order to obtain a pair (
2^{r1}*a*_{i}*b*_{j}^{2} = (1/2^{r2})Nm(*M*)

Now the norm of the element
Nm(*v*) = | *v*_{1}^{(i)} - *v*_{2}^{(i)}|^{ . }| - |^{2} 2^{r1}*a*_{i}×2^{2r2}*b*_{j}^{2}

Hence, we have the following

Nm(*v*) ^{ . }| *R*/*I*|

Here we have written |
While it is not too difficult to use this procedure to write all the
ideals *J* satisfying the above condition, it is much harder to write
the ``multiplication table'' for the group
Cl(*R*) on the basis of
what has gone so far. If *J*_{1} and *J*_{2} are two ideals as above and
the product no longer satisfies the above condition, then we need to
find the element *v* in
(*J*_{1}^{ . }*J*_{2}) that the lemma
guarantees. But the proof of the lemma gives us no way to find such
elements!