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Viewing as a Lattice
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Viewing as a Lattice
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The Volume of
Since
is a lattice in
, the volume of
is finite. Suppose
is a basis for
. Then if
is the matrix whose
th row is
, then we define the
volume
of
to be
.
Example
6
.
1
.
5
The ring
of integers of
has
-basis
,
. The map
is given by
The image
is spanned by
and
. The volume determinant is
Let
be the ring of integers of
. The map
is
and
which has determinant
, so the volume of
is
.
As the above example illustrates, the volume
need not be an integer.
William Stein 2012-09-24