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A place of a number field K, an object of type PlcNumElt,
is a class of absolute values (valuations) that
induce the same topology on the field. By a famous theorem of Ostrowski,
places of number fields are either finite, in which case they are in a
one-to-one correspondence with the on-zero prime ideals of the maximal order,
or infinite. The infinite places are identified with the embedding of K
into R or with pairs of embeddings into C.
The group of divisors is formally the free group generated by the finite
places and the R-vectorspace generated by the infinite ones.
Divisors are of type DivNumElt. Places have formal parent of
type PlcNum, while divisors belong to DivNum.
DivisorGroup(K) : FldNum -> DivNum
The set of places of the number field K and the group of divisors of K respectively.
d1 eq d2 : DivNum, DivNum -> BoolElt
p1 eq p2 : PlcNum, PlcNum -> BoolElt
NumberField(D) : DivNum -> FldNum
The number field for which P is the set of places or D is the group of divisors.
The place corresponding to prime ideal I.
Decomposition(K, I) : FldNum, Infty -> SeqEnum
A sequence of tuples of places and multiplicities. When a finite prime
(integer) 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.
For a number field K and a place p of the coefficient field of K,
compute all places (and their multiplicity) that extend p.
For finite places this is equivalent to the decomposition of the underlying
prime ideal. The sequence returned will contain the places of K extending
p and their ramification index.
For an infinite place p, this function will compute all extensions
of p in K. In this case, the integer returned in the second component
of the tuples will be 1 if p is complex or if p is real and extends to
a real place and 2 otherwise.
Decomposition(m, p) : Map[FldNum, FldNum], PlcNumElt -> SeqEnum[<PlcNumElt, RngIntElt>]
For an extension K/k of number fields (where k can be Q as well), given
by the embedding map m: k to K, decompose the place p of k in the
larger field.
In case k=Q, the place is given as either a prime number or zero to
indicate the infinite place.
The sequence returned contains pairs where the first component is a
place above p via m and the second is the ramification index.
InfinitePlaces(K) : FldRat -> SeqEnum
InfinitePlaces(K) : RngInt -> SeqEnum
A sequence containing all the infinite places of the number field K
is returned.
The divisor 1 * pl for a place pl.
The divisor which is the linear combination of the places corresponding to
the factorization of the ideal I and the exponents of that factorization.
The principal divisor xO where O is the maximal order of the
underlying number field of which x is an element. In particular, this computes a finite divisor.
RealPlaces(K) : FldNum -> [PlcNumElt]
For a number field K a sequence containing all real (infinite) places
is computed. For an absolute field this are precisely the embeddings into R
coming from the real roots of the defining polynomial.
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
Valuation(a, p) : RngElt, PlcNumElt -> RngElt
The valuation of the element a of a number field
at the place p.
The valuation of the ideal I at the finite place p.
The support of the divisor D as a sequence of places and a sequence
of the corresponding exponents.
Ideal(D) : PlcNumElt -> RngOrdIdl
The ideal corresponding to the finite part of the divisor D.
Evaluate(x, p) : FldRatElt, PlcNumElt -> RngElt
Evaluate(x, p) : RngIntElt, Infty -> RngElt
The evaluation of the number field element
x in the residue class field of the place p, i.e.
for a finite place p this corresponds to the image under
the residue class field map for the underlying prime ideal.
For infinite places, this returns the corresponding conjugate, ie.
a real or complex number.
The sequence of real embeddings of the algebraic number a is computed,
i.e. a is evaluated at all real places of the number field.
A sequence containing ∓ 1 depending on whether the evaluation
of the number field element a at the
corresponding real place is positive or negative.
For an infinite place p, returns true if the corresponding embedding
is real, i.e. if Evaluate at p will give real results.
For an infinite place p, return true if the corresponding embedding
is complex, i.e. if Evaluate at p will generally yield complex
results.
For a place p of a number field, return if the place is finite, i.e.
if it corresponds to a prime ideal.
For a place p of a number field return if the place is infinite,
ie. if it corresponds to an embedding of the number field into the
real or complex numbers. If the place is infinite, the index of the
embedding it corresponds to is returned as well.
For two places P of K and p of k where K is an
extension of k, check whether P extends p. For finite
places, this is equivalent to checking if the prime ideal corresponding
to P divides, in the maximal order of K the prime ideal of p.
For infinite places true implies that for elements of k,
evaluation at P and p will give identical results.
Degree(P) : PlcNumElt -> RngIntElt
For a place P of a number field, return the inertia degree of P.
That is for a finite place, return the degree of the residue class field
over it's prime field, for infinite places it is always 1.
For a divisor D of a number field, the degree is the weighted sum of the
degrees of the supporting places, the weights being the multiplicities.
NumberField(D) : DivNumElt -> FldNum
For a place P or divisor D of a number field, return the underlying number field.
For a place P of a number field, compute the residue class field of P.
For a finite place this will be a finite field, namely the
residue class field of the underlying prime ideal. For an
infinite place, the residue class field will be the field of
real or complex numbers.
For a finite place P of a number field, return an element of
valuation 1. This will be the uniformizing element of the
underlying prime ideal as well.
The degree of the completion at the place P, i.e. the product of the inertia degree
times the ramification index.
The ramification index of the place P. For infinite real places this is 1 and
2 for complex places.
For a place P of a normal number field, return the decomposition group
as a subgroup of the (abstract) automorphism group.
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