We say a field $F$ has the property $\*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $\*$. On the other hand if $ F $ is a finite field then by Chevalley-Warning theorem the equation $ x^2 + y^2 =-1 $ always has a solution.
I want to know which fields that satisfy property $ \* $. Does this type of field belong to a special class of field?



Fields in which $ -1 $ can't be written as sum of two square elements
In the notation of Lam's *Quadratic forms over fields*, the *Stufe* (a German word) or *level* (its English translation) $s(F)$ of a field is the minimal $n$ such that $-1$ is the sum of $n$ squares.


A theorem of Pfister says that $s(F)$ is always a power of $2$ (or $\infty$).


So you are taking of fields $F$ such that $s(F) > 2$ (or $s(F) \geq 4$, if you are willing to use Pfister's theorem).


