1. The $u$invariant problem
The problems in this section are related to the uinvariant problem on the maximal dimension of an anisotropic quadratic form.
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 1.2.
Find interesting classes $\mathcal F$ of fields such that for $F \in \mathcal F$, \[u(F) < \infty \Longrightarrow u(F(t)) < \infty.\]
\[ \mathcal F = \{F \ \ :\ \ F^{\times}/F^{\times 2} < \infty \} \] 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 $u_3(F)$? 
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 periodindex bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?
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?
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.