Pythagoras numbers of orders in biquadratic fields

Abstract

We examine the Pythagoras number P(OK) of the ring of integers OK in a totally real biquadratic number field K. We show that the known upper bound 7 is attained in a large and natural infinite family of such fields. In contrast, for almost all fields Q(5,s) we prove P(OK)=5. Further we show that 5 is a lower bound for all but seven fields K and 6 is a lower bound in an asymptotic sense.