A totally real field is a number field all of whose embeddings into 
have image in 
. A totally real field of
degree 
has 
such real embeddings, and
their joint map identifies its ring
of integers with a point lattice in 
(Milne 2020, pp. 79-80 and 90). For example, 
is totally real, with embeddings
sending 
to 
and 
. The field 
is not totally real.
Real embeddings of nonzero fractional ideals give point
lattices in 
. After normalization to covolume 1, their orbits under the group
of diagonal matrices with positive
diagonal entries and determinant 1 are compact.
These are periodic torus orbits occurring
in periodic torus packets (Einsiedler et al. 2011, pp. 823-824).
