Sums of Integral Squares In Certain Complex Bi-quadratic Fields (2005.13870v1)

Abstract: Let K be an algebraic number field and O_K be its ring of integers. Let S_K be the set of elements in O_K which are sums of squares in O_K and s(O_K) the minimal number of squares necessary to represent -1in O_K. Let g( S_K ) be the smallest positive integer t such that every element in S_K is a sum of t squares in O_K. Here K is generated over field of rational number by square root of m and -n , where m congruent 3 mod 4 and n congruent 1 mod 4 are two distinct positive square free integers, we prove that $ S_K= O_K. We also prove that g(O_ K) less or equals to s(O_K)+1 or s(O_K)+2. Applying this, we shows that if s(O_K)=2, then g(O_K)=3. This work is continuation of a recent study initiated by Zhang and Ji .