Places and Divisors of Number Fields

Subsections

• Creation of structures
• Creation of elements
• Arithmetic with places and divisors
• Other functions for Divisors and Places

Creation of structures

Places(K) : FldNum -> PlcNum

DivisorGroup(K) : FldNum -> DivNum

The set of places of K and the group of divisors of K respectively.

Creation of elements

Place(I) : RngOrdIdl -> PlcNumElt

The place corresponding to prime ideal I.

Decomposition(K, p) : FldNum, RngIntElt -> SeqEnum

Decomposition(K, I) : FldNum, Infty -> SeqEnum

A sequence of tuples of places and multiplicities. When finite prime p is given, the places and multiplicities correspond to the decomposition of p in the maximal order of K. When the infinite prime is given, a sequence of all infinite places is returned.

Divisor(pl) : PlcNumElt -> DivNumElt

The divisor 1 * pl.

Divisor(I) : RngOrdFracIdl -> DivNumElt

The divisor which is the linear combination of the places corresponding to the factorization of I and the exponents of that factorization.

Arithmetic with places and divisors

Divisors and places can be added, negated, subtracted and multiplied and divided by integers.

d1 + d2 : DivNumElt, DivNumElt -> DivNumElt

p + d : PlcNumElt, DivNumElt -> DivNumElt

d + p : DivNumElt, PlcNumElt -> DivNumElt

p1 + p2 : PlcNumElt, PlcNumElt -> DivNumElt

- p : PlcNumElt -> DivNumElt

- d : DivNumElt -> DivNumElt

d - p : DivNumElt, PlcNumElt -> DivNumElt

p - d : PlcNumElt, DivNumElt -> DivNumElt

d1 - d2 : DivNumElt, DivNumElt -> DivNumElt

p1 - p2 : PlcNumElt, PlcNumElt -> DivNumElt

p * k : PlcNumElt, RngIntElt -> DivNumElt

d * k : DivNumElt, RngIntElt -> DivNumElt

p div k : PlcNumElt, RngIntElt -> DivNumElt

d div k : DivNumElt, RngIntElt -> DivNumElt

Other functions for Divisors and Places

Valuation(a, p) : FldNumElt, PlcNumElt -> RngElt

Valuation(a, p) : RngOrdElt, PlcNumElt -> RngElt

The valuation of the element a of a number field or order at the place p.

Support(D) : DivNumElt -> SeqEnum, SeqEnum

The support of the divisor D as a sequence of places and a sequence of the corresponding exponents.