Let be a subfield of
, and assume that
is
separable. We say that a linear algebraic group
is defined
over
when its defining polynomial equations can be chosen
with coefficients in
. We can form the group
of
-rational points of
. In other words,
is the set of fixed points in
for the standard
action of the Galois group
on
.
The group
is called an
-form of
.
More generally, one defines
-forms
of
by
``twisting'' the standard action of the Galois group
on
by automorphisms of
.
Plain examples of -forms of reductive groups are
,
, the symplectic groups
, and the orthogonal
groups
. Depending on the properties of the field
,
there may exist many other forms of such ``classical'' matrix groups.
For instance,
and
are real forms of
the simple groups
and
respectively.
In addition, there are certain exceptional simple groups
that occur over any field .
A local field is a field equipped with a locally compact
non-discrete Hausdorff topology. The local fields of
characteristic
are
,
,
and finite extensions of the p-adic
field
.
The finite extensions of
are called p-adic fields
(of characteristic
).
Given a local field
, a form
of
-rational points of
an algebraic group
can be given the topology induced by the
topology of
. This gives
the structure of a locally compact
group. There is an important result saying that forms
of reductive algebraic groups over local fields are
always of type I. This statement provides a large class of examples
of type I groups for which the questions raised
in the previous section are meaningful.