\({\mathbb{Z}}\) 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.