Sums of Integral Squares In Certain Complex Bi-quadratic Fields

TL;DR

This paper investigates sums of squares in certain complex bi-quadratic fields, proving that all integers in the ring are sums of squares and establishing bounds on the minimal number of squares needed to represent elements.

Contribution

It demonstrates that in specific bi-quadratic fields, every element is a sum of squares and refines bounds on the minimal number of squares for representations.

Findings

All elements in the ring of integers are sums of squares.

Bounds on the minimal number of squares to represent elements are established.

If the minimal number for representing -1 is 2, then three squares suffice for all elements.

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 .