CyclotomicField(
n ) F
CyclotomicField(
gens ) F
CyclotomicField(
subfield,
n ) F
CyclotomicField(
subfield,
gens ) F
The first version creates the n-th cyclotomic field. The second version creates the cyclotomic field generated by gens. In both cases the field can be generated as an extension of a designated subfield.
AbelianNumberField(
n,
stab ) F
fixed field of the group generated by stab (prime residues modulo n) in the cyclotomic field with conductor n.
GaussianRationals V
is the field Q(i) of Gaussian rationals.
IsGaussianRationals(
obj ) C
ZumbroichBase(
n,
m ) F
is the set of exponents e for which E(
n)^
e belongs to the
(generalized) Zumbroich base of the cyclotomic field Qn,
viewed as vector space over Qm.
The base, the base conversion and the reduction to the minimal cyclotomic field are described in Zum89.
Note that for n º 2 mod 4 we have
ZumbroichBase(
n, 1 ) = 2 * ZumbroichBase(
n/2, 1 )
but
List( ZumbroichBase(
n, 1 ), x -> E(
n )^x ) =
List( ZumbroichBase(
n/2, 1 ), x -> E(
n/2 )^x )
.
LenstraBase(
n,
stabilizer,
super,
m ) F
is a list of lists of integers, each list indexing the exponents of an orbit of a subgroup of stabilizer on n-th roots of unity.
super is a list representing a supergroup of stabilizer which shall act consistently with the action of stabilizer, i.e., each orbit of supergroup is a union of orbits of stabilizer.
m is a positive integer. The basis described by the returned list is an integral basis over the cyclotomic field Qm.
Note that the elements are in general not sets, since the first element
is always an element of ZumbroichBase(
n,
m )
;
this property is used by NF
and Coefficients
.
Note that stabilizer must not contain the stabilizer of a proper cyclotomic subfield of the n-th cyclotomic field.
For details about the bases see Bre97.
IsNumberField(
F ) P
A number field is a finite extension of a prime field in characteristic zero.
IsAbelianNumberField(
F ) P
An abelian number field is a number field that is a Galois extension of the prime field with abelian Galois group (see GaloisGroup.field).
Conductor(
cyc ) A
Conductor(
C ) A
For an element cyc of a cyclotomic field, Conductor
returns the
smallest integer n such that cyc is contained in the n-th
cyclotomic field.
For a collection C of cyclotomics (for example a dense list of
cyclotomics or a field of cyclotomics), Conductor
returns the
smallest integer n such that all elements of C are contained in the
n-th cyclotomic field.
GaloisStabilizer(
F ) A
For an abelian number field F, GaloisStabilizer
returns
the set of all integers k in the range from 1 to the conductor of
F such that the field automorphism induced by raising roots of unity
in F to the k-th power acts trivially on F.
ComplexConjugate(
z ) A
For a cyclotomic number z, ComplexConjugate
returns
GaloisCyc(
z, -1 )
.
For a quaternion z = c1 e + c2 i + c3 j + c4 k,
ComplexConjugate
returns c1 e - c2 i - c3 j - c4 k.
GaussianIntegers V
is the ring of Gaussian integers. This is the subring Z[i] of the complex numbers, where i is a square root of -1.
IsGaussianIntegers(
obj ) C
is the defining category for the domain GaussianIntegers
.
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GAP 4 manual