Definition: Let $R$ be a commutative ring. The level of $R$, denoted $s(R)$, is the least positive integer $s$ such that $-1$ can be written as the sum of $s$ many squares in $R$. Set $s(R)=\infty$ if no such $s$ exists.
You might recall that a field $F$ is formally real if and only if $s(F)=\infty$.
Question: Let $F_n$ be the field of fractions of the integral domain $$\mathbb{Q}[x_1,\ldots,x_n]/(x_1^2+\ldots+x_n^2+1).$$ What is $s(F_n)?$ (It is clearly bounded above by $n$.)
$$~$$$$~$$
I have been reading some literature about this and the question of possible levels for fields was worked out by the following result of Pfister in 1965.
Theorem:
- If $F$ is a field with $s(F)<\infty$ then $s(F)$ is a power of 2.
- Let $m\geq 0$ and suppose $2^m\leq n<2^{m+1}$. Let $F=\mathbb{Q}(x_1,\ldots,x_n)$ and set $d=x_1^2+\ldots+x_n^2\in F$. Then $s(F(\sqrt{-d}))=2^m$.
So the powers of 2 are exactly the positive integers that can occur as the level of a field.
For general commutative rings there are more possibilities. In particular, there is this result of Dai, Lam, Peng from 1980.
Theorem: Let $R_n=\mathbb{Q}[x_1,\ldots,x_n]/(x_1^2+\ldots+x_n^2+1)$. Then $s(R_n)=n$.
The proof (at least the one I read) of that last one is much easier than Pfister's result, and is interesting in the sense that it uses the famous Borsuk-Ulam Theorem from topology.
So my question asks about the level of the field of fractions of $R_n$, which again is bounded above by $n$. But by Pfister's result it must be a power of 2. For example it should be easy to show $s(F_3)=2$. In light of part two of Pfister's theorem, maybe it's reasonable to expect $s(F_n)$ to be the greatest power of 2 that is still not more than $n$. Maybe there is a way of altering the proof of part two of Pfister's theorem to make it work for $F_n$.
The link I have been reading for all of this is the following: http://www.math.tifr.res.in/~anands/Level.pdf
No need, Pfister did it. This is Theorem 2.6 on page 382 of Lam's 2005 book, in particular formula (2.8) and the Remark partway down the page. Lam's book is reference [5] in the pdf you are reading.
=-=-=-=-=-=-=-=-=-=-=-=-=-=
=-=-=-=-=-=-=-=-=-=-=-=-=-=