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 $ * $. Does this type of field belong to a special class of field?
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1$\begingroup$ For what it's worth, these are precisely the fields over which the quaternion group $Q_8$ has a four dimensional irreducible representation. You might also want to look up the theory of Severi-Brauer varieties. $\endgroup$– Dave BensonCommented Jul 10, 2023 at 6:51
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$\begingroup$ "Additionally, it is assumed": you don't need this assumption, which is an immediate consequence of the definition. $\endgroup$– YCorCommented Jul 10, 2023 at 7:02
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$\begingroup$ Yes we don't need this assumption. I have edited. $\endgroup$– SkyCommented Jul 10, 2023 at 7:14
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2$\begingroup$ There are many such fields with rather disparate properties, and I don’t think there is any kind of classification. FWIW, these are called the fields of Stufe 2. $\endgroup$– Emil JeřábekCommented Jul 10, 2023 at 7:37
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1$\begingroup$ I mean, of Stufe more than $2$. $\endgroup$– Emil JeřábekCommented Jul 10, 2023 at 7:46
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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).
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2$\begingroup$ Why would you be not willing to use Pfister's theorem? In this special case, it is an immediate consequence of the Brahmagupta–Fibonacci identity, hence it is has a short, completely elementary proof. $\endgroup$ Commented Jul 10, 2023 at 8:26
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$\begingroup$ What will be the stuff for padic field? $\endgroup$– SkyCommented Jul 30, 2023 at 18:52