Pythagoras numbers for infinite algebraic fields

Abstract

We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.

11 pages; comments are welcome!

References in corpus (6)

• Lifting problem for universal quadratic forms
• Pythagoras numbers of orders in biquadratic fields
• A cubic ring of integers with the smallest Pythagoras number
• On the Pythagoras number of the simplest cubic fields
• Universal quadratic forms and Northcott property of infinite number fields
• Most totally real fields do not have universal forms or Northcott property