Let K be a number field of degree k > 1. We would like to know if a positive integer N can be represented as the sum, or the difference, of two norms of integral ideals of K. Suppose K/ℚ is abelian of conductor Δ. Then from the class field theory (Artin's reciprocity law) the norms are fully characterized by the residue classes modulo Δ. Precisely, a prime number p ∤ Δ (unramified in K) is a norm (splits completely in K), if, and only if,
where k is a subgroup of (ℤ/Δℤ)* of index k. Accordingly we may ask N to be represented as the sum
or the difference
of positive integers a, b each of which splits completely in K. For N to be represented in these ways the following congruences
must be solvable in α β є k, respectively. Moreover the condition
must hold. Presumably the above local conditions are sufficient for (−) to have infinitely many solutions and for (+) to have arbitrarily many solutions, provided N is sufficiently large in the latter case.