Consider the field of fractions $K$
of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$,
where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables.
Clearly $-1$ is a sum of 4 squares in $K$.
How can one prove that $-1$ is not a sum of 2 squares in $K$?


Serre mentions without proof this (probably known or easy) fact
in a letter to Eva Bayer of May 1, 2010,
and I am stuck: I cannot prove it.



Is -1 a sum of 2 squares in a certain field K?
This is a special case of a theorem of A. Pfister. It is well known to quadratic forms specialists. See e.g. Theorem XI.2.6 in T.Y. Lam's *Introduction to Quadratic Forms over Fields*. 


I believe the original paper is 



> 
> Pfister, Albrecht,
>  *Zur Darstellung von $-1$ als Summe von Quadraten in einem Körper*. (German)
>  J. London Math. Soc. 40 1965 159--165. 
> 
> 
> 


In this same paper Pfister defines the "stufe" (which Lam has successfully campaigned to be called the "level") of a non-formally real field, namely the least positive integer $n$ such that $-1$ is a sum of $n$ squares. Among his other achievements, he proves that the level is always a power of $2$ (so that Kevin Buzzard's recollection is correct). It is also worth remarking that his work is an insightful and rapid response to previous work of J.W.S. Cassels.


