Is -1 a sum of 2 squares in a certain field K? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:14:12Z http://mathoverflow.net/feeds/question/23608 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23608/is-1-a-sum-of-2-squares-in-a-certain-field-k Is -1 a sum of 2 squares in a certain field K? Mikhail Borovoi 2010-05-05T18:23:53Z 2010-06-22T14:21:22Z <p>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\$?</p> <p>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.</p> http://mathoverflow.net/questions/23608/is-1-a-sum-of-2-squares-in-a-certain-field-k/23609#23609 Answer by Pete L. Clark for Is -1 a sum of 2 squares in a certain field K? Pete L. Clark 2010-05-05T18:35:49Z 2010-06-22T14:21:22Z <p>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 <em>Introduction to Quadratic Forms over Fields</em>. </p> <p>I believe the original paper is </p> <blockquote> <p>Pfister, Albrecht, <em>Zur Darstellung von \$-1\$ als Summe von Quadraten in einem Körper</em>. (German) J. London Math. Soc. 40 1965 159--165. </p> </blockquote> <p>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.</p>