A cubic ring of integers with the smallest Pythagoras number

Abstract

We prove that the ring of integers in the totally real cubic subfield K-(49) of the cyclotomic field Q(zeta(7)) has Pythagoras number equal to 4. This is the smallest possible value for a totally real number field of odd degree.

Moreover, we determine which numbers are sums of integral squares in this field, and use this knowledge to construct a diagonal universal quadratic form in five variables.

Keywords

• cubic
• ring
• integers
• with
• smallest
• Pythagoras
• number