IsRootOfUnity(
cyc ) P
IsRootOfUnity
tests if a given cyclotomic is actually a root of unity.
CoeffList2CyclotomicList(
list,
root ) O
CoeffList2CyclogomicList(
list,
root )
takes a list of integers
list and a root of unity root and returns a list list2, where
list2[i]=list[i]* root^(i-1).
AbssquareInCyclotomics(
list,
root ) O
For a list of integers and a root of unity,
AbssquareInCyclotomics(
list,
root )
returns
the modulus of Sum(CoeffList2CyclotomicList(
list,
root ))
.
CycsGivenCoeffSum(
sum,
root ) O
CycsGivenCoeffSum(
sum,
root )
returns all elements of Z[ root ]
such that the coefficient sum is sum and all coefficients are
non-negative.
The returned list has the following form:
The cyclotomic numbers are represented by coefficients.
CoeffList2CyclotomicList
can be used to get the
algebraic number represented by list.
The list is partitioned into equivalence classes of elements having the
same modulus.
For each class the modulus is returned.
This means that CycsGivenCoeffSum
returns a list of pairs where the first
entry of each pair is the square of the modulus of an element of the
second entry. And the second entry is a list of coefficient lists of
cyclotomics in Z[ root ] having the coefficient sum sum.
gap> CycsGivenCoeffSum(3,E(3)); [ [ 0, [ [ 1, 1, 1 ] ] ], [ 3, [ [ 0, 1, 2 ], [ 0, 2, 1 ], [ 1, 0, 2 ], [ 1, 2, 0 ], [ 2, 0, 1 ], [ 2, 1, 0 ] ] ], [ 9, [ [ 0, 0, 3 ], [ 0, 3, 0 ], [ 3, 0, 0 ] ] ] ] gap> CycsGivenCoeffSum(2,E(2)); [ [ 0, [ [ 1, 1 ] ] ], [ 4, [ [ 0, 2 ], [ 2, 0 ] ] ] ]
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