Sums of three integral squares in biquadratic fields

TextLet K   be an algebraic number field and OKOK the ring of integers in K  . Let SKSK be the set of all elements α∈OKα∈OK which are sums of squares in OKOK and s(OK)s(OK) the minimal number of squares necessary to represent −1 in OKOK. Let g(SK)g(SK) be the smallest positive integer t   such that every element in SKSK is a sum of t   squares in OKOK. In this note, for K=Q(−m,−n), where m≡n≡3(mod4) are two distinct positive square-free integers, we show that SK=OKSK=OK and if s(OK)=2s(OK)=2, then g(OK)=3g(OK)=3.VideoFor a video summary of this paper, please click here or visit http://youtu.be/F6UHHgzfdzA.