MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Reading Lam's Introduction to Real Algebra, he remarks that:

  1. For a Dedekind domain $A$ with quotient field $F$, then $s(A)$ is either $s(F)$ or $s(F) + 1$. Furthermore, $s(A)$ is either $\infty$, $2^{n}$ or $2^{n} + 1$ for integers $n \geq 0$. For $n \geq 2$ there are no examples in literature of an $A$ such that $s(F) = 2^n$ and $s(A) = 2^n +1$.
  2. For a regular local ring, the relation between $s(A)$ and $s(F)$ is known(equality) for low dimensions, and he conjectures that this holds for higher dimensions.

As Lam's paper was written in 1984 I was hoping to find out whether there have been any development in respect to these two questions. I have tried searching for it, but haven't found anything related to it.

Background: The level of a commutative ring, denoted $s(A)$, is the smallest natural number $n$ such that $-1$ can be expressed as a sum of $n$ squares in $A$. if $-1$ is not a sum of squares we define the level of $A$ to be $\infty$.

share|cite|improve this question
up vote 1 down vote accepted

Concerning question 1, such examples have been found independently by David Leep (unpublished) and J. K. Arason and R. Baeza:

Arason, J. K.; Baeza, R. On the level of principal ideal domains. Arch. Math. (Basel) 96 (2011), no. 6, 519–524

share|cite|improve this answer
Thank you very much. – acyrl Nov 20 '12 at 20:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.