
## 1. The $u$-invariant problem

The problems in this section are related to the u-invariant problem on the maximal dimension of an anisotropic quadratic form.
1. #### Problem 1.1.

If $u(F) < \infty$, does there exist $B \in \N$ such that $u(F') < \infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$ sufficient?
• #### Problem 1.6.

When is $\tau_F$ finite?
• #### Problem 1.2.

Find interesting classes $\mathcal F$ of fields such that for $F \in \mathcal F$, $u(F) < \infty \Longrightarrow u(F(t)) < \infty.$
Examples to try:

$\mathcal F = \{F \ \ :\ \ |F^{\times}/F^{\times 2}| < \infty \}$ and $\mathcal F’ = \{F \ \ : \ \ F^{\times} = F^{\times 2} \}$
• #### Problem 1.7.

When $F = \R$ , we have $u(F) = \infty$ but $\tau_F < \infty$. Is there an example where this holds for $F$ not formally real?
•     Define $u_k(F) = \max \{ \dim (q)\ \ |\ \ q \in I^k \}.$ Then $u_0 = u$.

#### Problem 1.3.

Can we compute $u_k(F)$? How about $u_3(F)$?
Note that forms in $I^3$ have trivial Clifford invariant.
• #### Problem 1.8.

Find reasonable classes $\mathcal F$ of fields such that $u(F)$ is a power of 2 for all $F \in \mathcal F$. For these fields, assume $k$ is the period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?
1. Remark. I think I’ve seen this somewhere before
•     Let $\nu(F) = \max \{n\ |\ I^n \neq 0 \}.$

#### Problem 1.4.

If $\nu(F) < \infty$, is it true that $\nu(F') < \nu(F) +1$ for all $F'/F$ finite?
1. Remark. $2^{\nu(F)} < u(F)$
• Remark. [org.aimpl.user:c] This is related to the computation of Galois groups of quadratically closed fields.
•     For a quadratic form $q$ over $F$, we define $\textrm{splitting degree} = \min\{[L:F]\ :\ q_{L}\ \textrm{is a direct sum of hyperbolics}\}.$ The torsion index of $F$, denoted $\tau_F$ is the maximum splitting degree, taken over all even dimensional $q$.

#### Problem 1.5.

Is $\tau_F < 2^{(\frac{u(F)}{2} -1)}$?

Cite this as: AimPL: Deformation theory and the Brauer group, available at http://aimpl.org/deformationbrauer.