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\begin{document}

\title{Algebra 6.427}
\author{Michael Vaughan-Lee}
\date{June 2013}
\maketitle

Algebra 6.427 is a family of $p$ algebras of order $p^{6}$ which are
immediate descendants of algebra 5.45. This family has $p+1$ descendants of
order $p^{7}$ given by a two parameter family with parameters $x,y$, with
the isomorphism type depending on the value of $y^{2}-\omega x^{2}$ (Here,
as elsewhere, $\omega $ is a primitive element modulo $p$.)

This is essentially the same as in the descendants of 5.45. First we need
representative pairs $(x,y)$ giving the $(p-1)$ \emph{non-zero} values of $%
y^{2}-\omega x^{2}$. We get the $\frac{p-1}{2}$ distinct non-zero squares
modulo $p$ with parameters $(x,0)$ with $0<x\leq \frac{p-1}{2}$. To obtain
the non-squares, find $a$ such that $a^{2}-\omega $ is not a square modulo $p
$, and take parameters $(ay,y)$ for $0<y\leq \frac{p-1}{2}$. In the case $p=1%
\func{mod}4$, $a=0$ will do. I don't think the search for $a$ is linear in $p
$ for $p=3\func{mod}4$, but since $a^{2}-\omega $ is not a square modulo $p$
for half of the possible values of $a$, you would have to be unlucky not to
find a suitable $a$ quickly. We also need to find a single pair $(x,y)$ with 
$y^{2}-\omega x^{2}=\omega $, and we can find such a pair by evaluating $%
y^{2}-\omega (ay)^{2}$ for $0<y\leq \frac{p-1}{2}$.

\end{document}