call
as the multiplicative group of squares. so we decompose
.
is the set of non squares. suppose an element
is not expressible as sum of two squares. consider
. this set must be the same as
. consider
. clearly
but
was the coset of the multiplicative group
in
. so
. it follows that there exists a
s.t
RHS belongs to
whereas LHS belongs to
which are disjoint.