This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is [the Wikipedia explanation](https://en.wikipedia.org/wiki/U-invariant). In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin in 2001 who constructed a field with $u$-invariant of $9$.<sup>1</sup> > ***Q***. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis? As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number. <hr /> <sup>1</sup> "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" <a href="https://zbmath.org/?format=complete&q=an:0998.11015">ZBL</a>.