Maximal Orders


In Hecke, maximal orders (aka ring of integers), due to their special properties normal orders don't share, come with their own type NfMaximalOrder. While the elements have type NfOrderElem, the ideals and fractional ideals have types NfMaximalOrderIdeal and NfMaximalOrderFracIdeal respectively.

While theoretically a number field contains a unique maximal order (the set of all integral elements), for technical reasons in Hecke a number field admits multiple maximal orders, which are uniquely determined by the number field and a chosen integral basis.

Let $K$ be a number field of degree $d$ with primitive element $\alpha$ and $\mathcal O$ a maximal order $K$ with $\mathbf{Z}$-basis $(\omega_1,\dotsc,\omega_d)$. The basis matrix of $\mathcal O$ is the unique matrix $M_{\mathcal O} \in \operatorname{Mat}{d \times d}(\mathbf{Q})$ such that \begin{align} \begin{pmatrix} \omega_1 \\ \omega_2 \\ \vdots \\ \omega_d \end{pmatrix} = M{\mathcal O} \begin{pmatrix} 1 \\ \alpha \\ \vdots \\ \alpha^{d - 1} \end{pmatrix} \end{align}. If $\beta$ is an element of $\mathcal O$, we call the unique integers $(x_1,\dotsc,x_d) \in \mathbf Z^d$ with \begin{align} \beta = \sum_{i=1}^d x_i \omega_i \end{align} the coefficients of $\beta$ with respect to $\mathcal O$ or just the coefficient vector.

For an ideal $I$ of $\mathcal O$, a basis matrix of $I$ is a matrix $M \in \operatorname{Mat}{d \times d}(\mathbf{Z})$, such that the elements $(\alpha_1,\dotsc,\alpha_d)$ definied by \begin{align} \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_d \end{pmatrix} = M{\mathcal O} \begin{pmatrix} \omega_1 \\ \omega_2 \\ \vdots \\ \omega_d \end{pmatrix} \end{align} form a $\mathbf{Z}$-basis of $I$.