Sums of Integral Squares In Certain Complex Bi-quadratic Fields
Abstract
Let K be an algebraic number field and OK be its ring of integers. Let SK be the set of elements in OK which are sums of squares in OK and s(OK) the minimal number of squares necessary to represent -1in OK. Let g( SK ) be the smallest positive integer t such that every element in SK is a sum of t squares in OK. 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 $ SK= OK. We also prove that g(O K) less or equals to s(OK)+1 or s(OK)+2. Applying this, we shows that if s(OK)=2, then g(OK)=3. This work is continuation of a recent study initiated by Zhang and Ji .
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