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 archimedean 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. See here for more information on the different types of fields supported.

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 \[ K \longrightarrow \mathbf R^{d}, \ \alpha \longmapsto (\sigma1(\alpha),\dotsc,\sigmar(\alpha),\sqrt{2}\operatorname{Re}(\sigma{r+1}(\alpha)),\sqrt{2}\operatorname{Im}(\sigma{r+1}(\alpha)),\dotsc,\sqrt{2}\operatorname{Re}(\sigma{r+s}(\alpha)),\sqrt{2}\operatorname{Im}(\sigma{r+s}(\alpha)) \] 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

\[ \begin{pmatrix} \omega1 \\ \vdots \\ \omegad \end{pmatrix} = B \begin{pmatrix} 1 \\ \vdots \\ \alpha^{d - 1} \end{pmatrix} \]

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 \[ \begin{pmatrix} \sigma1(\omega1) & \dotsc & \sigmar(\omega1) & \sqrt{2}\operatorname{Re}(\sigma{r+1}(\omega1)) & \sqrt{2}\operatorname{Im}(\sigma{r+1}(\omega1)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma{r+s}(\omega1)) \\ \sigma1(\omega2) & \dotsc & \sigmar(\omega2) & \sqrt{2}\operatorname{Re}(\sigma{r+1}(\omega2)) & \sqrt{2}\operatorname{Im}(\sigma{r+1}(\omega2)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma{r+s}(\omega2)) \\ \vdots & \dotsc & \vdots & \vdots & \dotsc & \vdots & \vdots\\ \sigma1(\omegad) & \dotsc & \sigmar(\omegad) & \sqrt{2}\operatorname{Re}(\sigma{r+1}(\omegad)) & \sqrt{2}\operatorname{Im}(\sigma{r+2}(\omegad)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma{r+s}(\omegad)) \end{pmatrix} \in \operatorname{Mat}_{d\times d}(\mathbf R). \] is called the Minkowski matrix of $\mathcal O$.

Examples

Usually, to create an order, one starts with a field (or a polynomial):

julia> Qx, x = PolynomialRing(FlintQQ, "x");

julia> K, a = NumberField(x^2 - 10, "a");

julia> E = EquationOrder(K)
Order of Number field over Rational Field with defining polynomial x^2-10
with Z-basis NfAbsOrdElem{AnticNumberField,nf_elem}[1, a]

julia> Z_K = MaximalOrder(K)
Maximal order of Number field over Rational Field with defining polynomial x^2-10
with basis nf_elem[1, a]

julia> conductor(E)
Ideal of (Order of Number field over Rational Field with defining polynomial x^2-10
with Z-basis NfAbsOrdElem{AnticNumberField,nf_elem}[1, a])
with basis matrix
[1 0]
[0 1]

julia> E == Z_K
true

Once orders are created, we can play with elements and ideals:

julia> lp = prime_decomposition(Z_K, 2)
1-element Array{Tuple{NfAbsOrdIdl{AnticNumberField,nf_elem},Int64},1}:
 (<2, a>
Norm: 2
Minimum: 2
two normal wrt: 2, 2)

julia> p = lp[1][1]
<2, a>
Norm: 2
Minimum: 2
two normal wrt: 2

julia> isprincipal(p)
(false, 1)

julia> fl, alpha = isprincipal(p^2)
(true, -2)

julia> norm(alpha)
4

It is possible to work with residue fields as well:

julia> Fp, mFp = ResidueField(Z_K, p)
(Finite field of degree 1 over F_2, Map with following data
Domain:
=======
Maximal order of Number field over Rational Field with defining polynomial x^2-10
with basis nf_elem[1, a]

Codomain:
=========
Finite field of degree 1 over F_2
)

julia> [ mFp(x) for x = basis(Z_K)]
2-element Array{fq,1}:
 1
 0