$-1$ as sum of $k$ squares with $k$ least postive integer Let $F$ be a finite extension of $\mathbb{Q}$ inside $\mathbb{C}$ such that it do not contains $i$. 
Let $k$ be the smallest positive integer such that $-1$ is sum of $k$ squares in $F$.
A theorem of Kaplansky (1953) asserts that $k$ is either $1,2,4,8$ or a multiple of $8$. Kaplansky also points out in his corresponding paper that $k=16$ will not occur.
Since this paper is more than 50 year old, it seems that there may be further development related to the above fact. This raised a question to me:
Question. For $F$ as above, is there now complete determination of all possible value of $k$?

It should be noted that when field has characteristic $p$, then $k$ is either $1$ or $2$; hence consider fields of only characteristic zero; among them, I am considering more natural and specific fields - finite extension of $\mathbb{Q}$ which do not contain $i$ and ask for all possible values of $k$ whether estimated till today? If yes, can one suggest some reference for it, possibly a book?
 A: Your question makes sense for any field $F$, for which the level, denoted by $s(F)$ (from the German Stufe), is defined as the least positive integer $k$ such that $-1$ is a sum of $k$ squares in $F^*$ whenever such an integer
exists, and $\infty$  otherwise. For fields of positive characteristic, $s(F)$  can take only the values $1$ and $2$, depending just on whether $-1$ is a square in $F^*$ or not. Fields of level $\infty$, called real fields, are exactly those on which an ordering can be defined (e.g. $\mathbf Q$ or $\mathbf R$). Fields of finite level are also called nonreal fields.
The complete solution to the problem of the possible values of $s(F)$ was given by A. Pfister at the beginning of the $60$s, inspiring a big part of later advances in the theory of quadratic forms, e.g. the developments around the so called Pfister forms, see T.Y. Lam's book "The algebraic theory of quadratic forms", Benjamin (1973). Pfister proved that the level of a nonreal field is always a power of $2$, and further that, if $F$ is any real field  and $n\ge 0$, then the function field of the projective quadric $X_0^2 + ... +X_{2^n}^2 = 0$ over $F$ has level $2^n$. These were the first examples of nonreal fields of level $\ge 4$ and, actually, still no examples of an essentially different kind are known. The standard examples of fields of level $1,2$ and $4$, are respectively $\mathbf C, \mathbf F_3$ and $\mathbf Q_2$. But note that it remains a diffcult problem to determine the level of a given field of characteristic zero. For an overview on what is known about levels of common types of fields, see op. cit., chapter XI, section 2 .
