# Solving linear systems for integer values in MAGMA

Say we are given a quaternion algebra D over a number field F as well as a maximal $\mathcal{O}_F$-order $\Delta$ $\subseteq$ D and say we have a $\mathbb{Z}$-basis $\omega_1, . . . , \omega_n$ for $\Delta$ such that $\omega_1 = 1$.

Let $\mathfrak{b}$ be a fractional left $\Delta$-ideal. According to the definition of fractional ideals, we can find d $\in$ $\Delta$ such that d$\mathfrak{b}$ is an integral left ideal of $\Delta$. Hence we can find a $\mathbb{Z}$-basis $\omega'_1$, . . . , $\omega'_k$ of d$\mathfrak{b}$. Using MAGMA, I would now like to find a$_{ij}$ $\in$ $\mathbb{Z}$, i = 1,..,n, such that $\sum_{i=1}^{n} a_{ij} \omega_{i}$ = $\omega'_j$ for each j = 1,...,k. How can I compute these a$_{ij}$ $\in$ $\mathbb{Z}$ (!) using MAGMA? Using the Solution() function, I can compute coefficients in F but I need them to be integers.

• Check out the documentation for dealing with modules over Dedekind domains magma.maths.usyd.edu.au/magma/handbook/text/587 – Morgan Rodgers May 31 '18 at 16:42
• You should be able to view $\Delta$ as a module over $\mathbb{Z}$, then there is a function IsSubmodule() that will let you know if $\mathrm{d}\mathfrak{b}$ is a submodule of $\Delta$ (probably the trivial part) and if so, will return an embedding (which should allow you to represent your $\omega^{\prime}$ values in terms of your $\mathbb{Z}$-basis for $\Delta$). – Morgan Rodgers May 31 '18 at 16:45