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 $\infty$.


2 Answers 2


Concerning question 1, such examples have been found independently by David Leep (unpublished) and J. K. Arason and R. Baeza: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



Let me assume that $2$ is invertible in $A$.

Regarding question 2, for a regular local ring $A$ with fraction field $F$ and $s(F)=2^k$, the equality $s(A)=s(F)$ follows from the Gorthendieck--Serre conjecture for the (split) group scheme $\mathrm{SO}(2^{k+1})$ over $A$. This case of the conjecture is known for many regular local rings, e.g., when $A$ contains a field (see below), but is open in general.

The Grothendieck--Serre conjecture states that if $A$ is a regular local ring with fraction field $F$ and $G$ is a reductive group scheme over $A$, then the base-change map of pointed etale cohomology sets $\mathrm{H}^1(A,G)\to \mathrm{H}^1(F,G)$ has trivial kernel. In the special case where $G$ is $\mathrm{SO}(2n)$, i.e., the group of isometries of a $2n$-dimensional hyperbolic quadratic form over $A$, the set $\mathrm{H}^1(A,G)$ is in bijection with isomorphism classes of (regular) $2n$-dimensional quadratic forms with trivial discriminant over $A$, so in this case, the conjecture says that a quadratic form over $A$ that becomes hyperbolic over $F$ is already hyperbolic over $A$.

Suppose that the Grothendieck-Serre conjecture holds for $\mathrm{SO}(2^{k+1})$ and consider the Pfister quadratic form $q:=2^{k+1}\times \langle 1\rangle =\langle\langle 1\rangle\rangle^{\otimes (k+1)}$ over $A$. Then $q_F$ is isotropic, hence hyperbolic, so our assumption implies that $q\cong 2^{k}\times \langle 1,-1\rangle$. It is known that Witt's Cancellation Theorem holds over local rings, so this means that $2^k\times\langle 1\rangle\cong 2^k\times \langle -1\rangle$ as quadratic forms over $A$. The right hand side represents $-1$, so $-1$ is a sum of $2^k$ squares in $A$, i.e., $s(A)\leq 2^k$. As $s(A)\geq s(F)$, we conclude that $s(A)=2^k$.

The general Grothendieck-Serre conjecture is known to hold in many cases. Most notably, it holds when $A$ contains a field; this is due to Fedorov-Panin and Panin. To my best knowledge, what is presently known for the particular group $\mathrm{SO}(2n)$ is that it satisfies the conjecture when:

  • $A$ is unramified, i.e., $A/pA$ is regular with $p$ be the characteristic of the residue field of $A$ (this includes the case where $A$ contains a field), or

  • $A$ has Krull dimension at most $4$.

The first case is due to Cesnavicius (who proved the conjecture for quasi-split groups over unramified local rings), and the second follows from the work of Balmer and Walter on the Gersten-Witt complex, which implies in particular that the base-change map between the Witt groups $W(A)\to W(K)$ is injective when $A$ is regular local of Krull-dimension $\leq 4$.


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