# ${\mathbb{Z}}$ as a Number Field Order

A collection of functions are provided that make ${\mathbb{Z}}$ behave like an order of a number field. Note however, that ${\mathbb{Z}}$ is not of type `RngOrd`. If complete compatibility is necessary, the user should create the maximal order of a degree $1$ extension of ${\mathbb{Q}}$.

## `Decomposition(R, p): RngInt, RngIntElt -> SeqEnum`

Returns the ideal decomposition of the prime $p$, i.e. a list `[ < ideal<Z|p>, 1> ]` as in the number field case.

## `Generator(I): RngInt -> RngIntElt`

A generator for the given ideal.

## `RamificationIndex(I, p): RngInt, RngIntElt -> RngIntElt`

## `RamificationIndex(I): RngInt -> RngIntElt`

The ramification index of $I$ over ${\mathbb{Z}}$ which is always $1$.

## `Degree(I): RngInt -> RngIntElt`

The inertia degree of the ideal $I$, which is always $1$.

## `TwoElementNormal(I): RngInt -> RngIntElt, RngIntElt`

Two integers that generate the ideal $I$. In this case the generator is returned twice.

## `ChineseRemainderTheorem(I, J, a, b): RngInt, RngInt, RngIntElt, RngIntElt -> RngIntElt`

The Chinese remainder theorem for ideals. Given ideals $I$ and $J$ of ${\mathbb{Z}}$ together with integers $a$ and $b$, an integer $x$ such that $x-a \in I$ and $x-b \in J$ is returned.

## `Valuation(x, I): RngIntElt, RngInt -> RngIntElt`

The valuation of the integer $x$ at the prime ideal $I$.

## `ClassRepresentative(I): RngInt -> RngInt`

The representative of the ideal $I$ of ${\mathbb{Z}}$ in the basis of the class group.