Pythagoras numbers of orders in biquadratic fields

Abstract

We examine the Pythagoras number of the ring of integers in a totally real biquadratic number field

. 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 we prove

. Further we show that 5 is a lower bound for all but seven fields and 6 is a lower bound in an asymptotic sense.

Keywords

• Pythagoras
• numbers
• orders
• biquadratic
• fields