A well-known open problem asks for the fewest number of colors necessary
to color the points of the plane so that any two points at distance 1
have different colors.
If one considers only points with rational coordinates,
then 2 colors suffice.
Several authors have considered
the problem of coloring all points whose coordinates lie in some real
quadratic field, or more generally, a subfield of the real numbers.
Fischer  describes "additive colorings" as follows.
Firstly, let G0(K)
be the connected component of the underlying graph,
whose vertices are points in K² ,
with an edge between two points at distance 1 .
Then an additive coloring is a homomorphism
f : G0(K) ---> A ,
where A is a finite abelian group.
As a result, the connectedness of the underlying graph becomes relevant.
In this paper, we show that if K is a number field,
then the graph on K² is connected
if and only if -1 is not a square in K .
We also consider higher dimensions, and show that the graph on
K³ is connected if and only if the graph on
K4 is connected, and give necessary and
sufficient conditions for this, in terms of the splitting of the
rational prime 2 in K .
Lastly, we show that the graph on Kd
is connected for all K when d ≥ 5 .