this is a question about quaternions, from someone whose background is graph theory but who knows only a little number theory.
- Let $\xi$ be a quaternion $\xi = a + b{\bf{i}} + c{\bf{j}} +d{\bf{ij}}$, in the field ${\mathbb{F}}_2[x]$; $\mathbb{F}_2$ is the field with 2 elements; where here:
--multiplication is defined: ${\bf{i}}^2 = \bf{i}+1$; ${\bf{j}}^2 = x$; ${\bf{ji}} = {\bf{ij}} +{\bf{j}}$.
--the norm $N(\xi)$ is defined here $N(\xi) = (a^2+ab+b^2) + (c^2+cd +d^2)x$.
Now define $\Lambda(x)$ to be the set of quaternions $\xi$ in the field ${\mathbb{F}}_2[x]$ as specified above such that $N(\xi) = (x+1)^r$ for some nonnegative integer $r$.
Now let $h(x)$ be an irreducible polynomial of degree 4 in ${\mathbb{F}}_2[x]$.
Now, I would like to know if the italicized statement is known to be true, and if so, is there a result in the literature you can point me:
For some integers $r$ and $l$, let $a$, $b$, $c$ and $d$ be polynomials in $\mathbb{F}_2[x]/h^l(x)$ such that the following equation $(a^2+ab+b^2) + (c^2+cd +d^2)x \equiv (x+1)^r$ mod $h^l(x)$ is satisfied. Then there exists a $\xi \in \Lambda(x)$ (i.e., $N(\xi) = (x+1)^{r'}$ for some nonegative integer $r'$ which may or may not be $r$) such that $\xi$ mod $h^l(x)$ $\equiv$ $a + b{\bf{i}} + c{\bf{j}} +d{\bf{ij}}$.
Many thanks for considering this. I have tried looking this up in the literature, I must admit, I do not have much of a background in number theory, so I have trouble following.
--Mike