# Level of a commutative ring and its quotient field.

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 $\infty$.

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Concerning question 1, such examples have been found independently by David Leep (unpublished) and J. K. Arason and R. Baeza: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