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? Additionally, it is assumed that the imaginary unit $i$ is not an element of $F$.

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?

Let a field $ F $ has the property $*$ if the equation $ x^2 + y^2=-1 $ has no solution. For an example if $ F $ is a subfield of real numbers then $ F $ satiesfied $ * $. On the other hand if $ F $ is finite field then by chevally-warning theorem the equation $ x^2 + y^2 =-1 $ has always a solution. I want to know for which field satisfied the property $ * $. Is that type of field belonging to a Special class of field.Additionally, it is assumed that the imaginary unit $i$ is not an element of $F$.

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? Additionally, it is assumed that the imaginary unit $i$ is not an element of $F$.