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I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true when the Pythagoras number of $K$ is $1$, and clearly false when it is $3$, but I am not sure what can happen when it is $2$. The two properties I am looking at are:

  1. The torsion subgroup of $W(K)$ has exponent (at most) $2$.
  2. For any $a,b\in K^\times$ which are sums of squares, the Pfister form $\langle\langle a,b\rangle\rangle$ is hyperbolic.

When $K$ is Pythagorean, 1. is true because $W(K)$ is either $\mathbb{Z}/2\mathbb{Z}$ or is torsion-free, and 2. is obvious since $a$ and $b$ are actually squares by hypothesis.

When the Pythagoras number is at least $3$, we can choose some $a\in K^\times$ which is a sum of squares, but not of $2$ squares. Then $\langle \langle a,a\rangle\rangle = \langle \langle -1,a\rangle\rangle$ is not hyperbolic, which already shows that 2. is false. But $a$ is positive in any real closure of $K$, so $\langle \langle a\rangle\rangle$ becomes hyperbolic in any real closure, and thus by Pfister's local-global theorem $\langle \langle a\rangle\rangle$ is a torsion element in $W(K)$. Since $2\langle \langle a\rangle\rangle = \langle \langle -1,a\rangle\rangle\neq 0$ in $W(K)$, this shows that 1. is also false.

Are those properties [true/false/it depends] when the Pythagoras number is $2$?

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1 Answer 1

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Let $p(K)$ be the Pythagoras number of $K$. Suppose that $p(K) = 2$.

If $K$ is a formally real field, then the torsion subgroup of $W(K)$ has exponent 2. If $K$ is a nonreal field, then this is false. For example, let $K$ be the finite field with $q$ elements where $q \equiv 3 \bmod 4$. Then $p(K) = 2$, but $W(K)$ has exponent 4 because $<1>$ has order 4 in $W(K)$. These results can be found in Lam's textbook on quadratic forms.

Statement (2) is false in general when $p(K) = 2$, but this is more difficult to show. Here are two examples. $K = \mathbb{R}((x,y))$ and $K = \mathbb{R}((x))(y)$. Proofs of these statements can be found in K.J. Becher, D.B. Leep, The length and other invariants of a real field, Math. Zeit., Volume 269 (2011), 235-252. See Example 6.6.

Some additional explanation is needed to understand this reference. Statement (2) of the question is equivalent to the condition $(I_t(K))^2 = 0$ where $I_t(K)$ is the set of torsion elements in the ideal of even dimensional forms in $W(K)$. (Note that $(I_t(K))^n$ is not the same as the more commonly studied object $I_t^n(K)$, which is the torsion part of $(IK)^n$.) It is shown in Proposition 4.6 of this paper that $\ell(K) \le 2$ if and only if $(I_t(K))^2 = 0$ where $\ell(K)$ denotes the length of the field, which is defined in section 4. This explanation of some of the basic notations and terminology of this paper will make it easier to understand Example 6.6 mentioned above.

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