Introduction
This chapter deals with absolute number fields and orders there of.
Definitions and vocabulary
We begin by collecting the necessary definitions and vocabulary. This is in particular important for everything related to embeddings of number fields into archimdean fields, since they are at least two (slighlty) different normalizations.
Number fields
By an absolute number field we mean finite extensions of $\mathbf Q$, which is
of type AnticNumberField and whose elements are of type nf_elem. Such an
absolute number field $K$ is always given in the form $K = \mathbf Q(\alpha) =
\mathbf Q[X]/(f)$, where $f \in \mathbf Q[X]$ is an irreducible polynomial. We
call $(1,\alpha,\alpha^2,\dotsc,\alpha^{d-1})$, where $d$ is the degree $[K :
\mathbf Q]$ the power basis of $K$. If $\beta$ is any element of $K$, then
the representation matrix of $\beta$ is the matrix representing $K \to K,
\gamma \mapsto \beta \gamma$ with respect to the power basis, that is,
\[ \beta \cdot (1,\alpha,\dotsc,\alpha^{d-1}) = M_\alpha (1, \alpha, \dotsc, \alpha^{d-1}). \]
Let $(r,s)$ be the signature of $K$, that is, $K$ has $r$ real embeddings $\sigma_i \colon K \to \mathbf{R}$, $1 \leq i \leq r$, and $2s$ complex embeddings $\sigma_i \colon K \to \mathbf{C}$, $1 \leq i \leq 2s$. In Hecke the complex embeddings are always ordered such that $\sigma_i = \overline{\sigma_{i+s}}$ for $r + 1 \leq i \leq r + s$. The $\mathbf{Q}$-linear function \[ K \longrightarrow \mathbf R^{d}, \ \alpha \longmapsto (\sigma_1(\alpha),\dotsc,\sigma_r(\alpha),\sqrt{2}\operatorname{Re}(\sigma_{r+1}(\alpha)),\sqrt{2}\operatorname{Im}(\sigma_{r+1}(\alpha)),\dotsc,\sqrt{2}\operatorname{Re}(\sigma_{r+s}(\alpha)),\sqrt{2}\operatorname{Im}(\sigma_{r+s}(\alpha)) \] is called the Minkowski map (or Minkowski embedding).
Orders
If $K = \mathbf Q(\alpha)$ is an absolute number field, then an order $\mathcal O$ of $K$ is a subring of the ring of integers $\mathcal O_K$, which is free of rank $[ K : \mathbf Q]$ as a $\mathbf Z$-module. The natural order $\mathbf Z[\alpha]$ is called the equation order of $K$. In Hecke orders of absolute number fields are constructed (implicitely) by specifying a $\mathbf Z$-basis, which is refered to as the basis of $\mathcal O$. If $(\omega_1,\dotsc,\omega_d)$ is the basis of $\mathcal O$, then the matrix $B \in \operatorname{Mat}_{d \times d}(\mathbf Q)$ with
\[ \begin{pmatrix} \omega_1 \\ \vdots \\ \omega_d \end{pmatrix} = B \begin{pmatrix} 1 \\ \vdots \\ \alpha^{d - 1} \end{pmatrix} \]
is called the basis matrix of $\mathcal O$. We call $\det(B)$ the generalized index of $\mathcal O$. In case $\mathbf Z[\alpha] \subseteq \mathcal O$, the determinant $\det(B)^{-1}$ is in fact equal to $[ \mathcal O : \mathbf Z[\alpha]]$ and is called the index of $\mathcal O$. The matrix \[ \begin{pmatrix} \sigma_1(\omega_1) & \dotsc & \sigma_r(\omega_1) & \sqrt{2}\operatorname{Re}(\sigma_{r+1}(\omega_1)) & \sqrt{2}\operatorname{Im}(\sigma_{r+1}(\omega_1)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma_{r+s}(\omega_1)) \\ \sigma_1(\omega_2) & \dotsc & \sigma_r(\omega_2) & \sqrt{2}\operatorname{Re}(\sigma_{r+1}(\omega_2)) & \sqrt{2}\operatorname{Im}(\sigma_{r+1}(\omega_2)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma_{r+s}(\omega_2)) \\ \vdots & \dotsc & \vdots & \vdots & \dotsc & \vdots & \vdots\\ \sigma_1(\omega_d) & \dotsc & \sigma_r(\omega_d) & \sqrt{2}\operatorname{Re}(\sigma_{r+1}(\omega_d)) & \sqrt{2}\operatorname{Im}(\sigma_{r+2}(\omega_d)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma_{r+s}(\omega_d)) \end{pmatrix} \in \operatorname{Mat}_{d\times d}(\mathbf R). \] is called the Minkowski matrix of $\mathcal O$.