(Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:19:13Z http://mathoverflow.net/feeds/question/73210 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73210/non-existence-of-skew-fields-satisfying-a-sgpi-skew-generalized-polynomial-id (Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity) Max 2011-08-19T09:26:58Z 2011-08-26T17:23:23Z <p>Let $K$ be a skew-field, infinite dimensional over its center $F$. </p> <p>From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-algebras have finite dimension over their center).</p> <p>There, a GPI (generalized polynomial identity) has coefficients from the center $F$. If the coefficients are arbitrary, one has a GI (generalized identity), and a theorem by Amitsur describing the possible structure of $K$.</p> <p>In my research, I now came upon skew-fields which might satisfy a "skew" GPI in the following sense: Let $\sigma$ be an involutory antiautomorphism of $K$ and $\gamma$ an automorphism of $K$ of order 1 or 2. For simplicity, let's restrict to the case that $\sigma$ and $\gamma$ commute.</p> <blockquote> <p>Is it possible that $K$ satisfy an "skew" GPI, meaning that coefficients are arbitrary from $K$, and the GPI contains not just $x$, but also $x^\gamma$, $x^\sigma$ and $x^{\gamma\sigma}$ ? The case with a single unknown is all that interests me.</p> </blockquote> <p>I actually kinda hope these skew fields don't exist respectively must be finite dimensional over their center. </p>