cyclotomic.at

CyclotomicField type

set_type CyclotomicField = (int order, [int] Phi)

Fields: order, Phi

CyclotomicFieldElement type

set_type CyclotomicFieldElement = ([rat] v,CyclotomicField F)

Fields: v, F

to_denominator

to_denominator(r,N): integer $a$ such that rational $r$ is congruent to
$a/N$ modulo 1; it is a precondition that |denom(r)| divides |N|.
The application is that $\exp(2\pi i r)=\zeta^a$ where $\zeta=\exp(2\pi i/N)$

cyclotomic_field

construct a cyclotomic field.

CF

abbreviation

degree

also defined in polynomial.at, character_tables.at, laurentPolynomial.at

=

also in this file at line 69, line 74; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, disconnected.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at

reduce_mod

a utility function like |%@(rat_poly,rat_poly)|, but fixing dividend first,
reducing |rat_poly| mod |int_poly|, and returning a remainder of fixed length

element

make constructor function for cyclotomic field elements, ensures reduction

cyclotomic_field_element

same constructor, but used with field as second argument

embed

embed Q in cyclotomic field as (images of) constant polynomials.

also defined in character_table_reps.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at

one

zero

=

also in this file at line 42, line 74; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, disconnected.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at

!=

also in this file at line 76; also defined in basic.at, combinatorics.at, synthetic.at, hash.at, tits.at, polynomial.at, cyclotomicMat.at

=

also in this file at line 42, line 69; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, disconnected.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at

!=

also in this file at line 73; also defined in basic.at, combinatorics.at, synthetic.at, hash.at, tits.at, polynomial.at, cyclotomicMat.at

primitive_root

+

additive operators do not need to call |reduce_by|

also defined in basic.at, extParamPol.at, modules.at, ratmat.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, hodge_functions.at, lazy_lists.at

- 2 overloads

negation and scalar multiplication need no reduction either

also defined in basic.at, extParamPol.at, modules.at, ratmat.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomicMat.at, hodge_functions.at, lazy_lists.at

* 2 overloads

cyclotomic field multiplication.

also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at, stable.at, synthetic.at, Weylgroup.at, group_operations.at, ratmat.at, tits.at, twisted_root_datum.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

is_rational

also defined in cyclotomicMat.at

to_rational

also defined in cyclotomicMat.at

is_integer

also defined in basic.at

to_integer

exp_1

embed $\Q$ as roots of unity in F via $t\mapsto\exp(2\pi i t)$

zeta_power

Galois_matrix

the Galois group over $\Q$ of |F| is the multiplicative group mod |F.order|
the function |Galois_matrix| gives the matrix on the standard basis (the
initial powers of |zeta|) of the action of the Galois group element for |k|

|gcd(k,F.order)=1|

Galois_matrices

Galois_action

matrix_of_multiplication

norm

inverse_by_norm

inverse of z=products of nontrivial Galois conjugates/norm.

inverse_by_matrix

inverse of z = inverse of (matrix of multiplication by z) applied to F.one.

inverse_by_solve

inverse

inverse of element in cyclotomic field.

also defined in basic.at, combinatorics.at, ratmat.at, parameters.at, tits.at, affine.at, Gaussian_elim.at, cyclotomic_Gaussian_elim.at, lazy_lists.at

/ 2 overloads

unary division is inverse

also defined in basic.at, ratmat.at, polynomial.at, lazy_lists.at

^

also defined in basic.at, ratmat.at, tits.at, polynomial.at, sparse.at, lazy_lists.at

order

order of element of F, returns 0 if element not of finite order

also defined in basic.at, all_finite_order.at, tits.at, extended.at

power_of_primitive_root

if z\in CF(n) is an n^th root of unity, write z=zeta_n^k

root_of_unity

in CF(n): if r=a/b and b divides n then return zeta_n^a

format

convert cyclotomic field element to a string for output.

also defined in polynomial.at, extended_misc.at, cyclotomicMat.at, bracket.at

show

display cyclotomic field element as polynomial in x.

also defined in modules.at, K_Nilpotent.at, good_W_representatives.at, arthur_parameters.at, sub_cells.at, associated_variety_annihilator.at, geck_generic.at, L_packet.at, ratmat.at, Levi_subgroups.at, nilpotent_orbits.at, subgroups.at, poset.at, polynomial.at, character_tables.at, springer_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, weak_packets.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, exceptional.at, nilpotent_posets.at, lazy_lists.at

complex_conjugate

bar

also defined in cyclotomicMat.at

is_real

also defined in basic.at, cyclotomicMat.at

absolute_value_squared

cyclotomic_factors

all cyclotomic factors of polynomial P, as list of positive integers

product_of_cyclotomic

write a rational polynomial as a product of cyclotomic polynomials, assumed to
be possible (if not we hang because it is hard to tell when to give up trying)
return value [a_1,...] means Phi_{a_1}*Phi_{a_2}*..