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Standard Commutative Rings
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Z
_
__pow__() (sage.rings.integer.Integer method)
A
absolute_degree() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
absolute_discriminant() (sage.rings.rational_field.RationalField method)
absolute_norm() (sage.rings.rational.Rational method)
additive_order() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
algdep() (in module sage.arith.misc)
algebraic_closure() (sage.rings.rational_field.RationalField method)
algebraic_dependency() (in module sage.arith.misc)
arith_int (class in sage.rings.fast_arith)
arith_llong (class in sage.rings.fast_arith)
aurifeuillian() (in module sage.rings.factorint)
automorphisms() (sage.rings.rational_field.RationalField method)
B
bernmm_bern_modp() (in module sage.rings.bernmm)
bernmm_bern_rat() (in module sage.rings.bernmm)
bernoulli() (in module sage.arith.misc)
bernoulli_mod_p() (in module sage.rings.bernoulli_mod_p)
bernoulli_mod_p_single() (in module sage.rings.bernoulli_mod_p)
binary() (sage.rings.integer.Integer method)
binomial() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
binomial_coefficients() (in module sage.arith.misc)
bits() (sage.rings.integer.Integer method)
C
ceil() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
characteristic() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
charpoly() (sage.rings.rational.Rational method)
class_number() (sage.rings.integer.Integer method)
(sage.rings.rational_field.RationalField method)
completion() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
complex_embedding() (sage.rings.rational_field.RationalField method)
conjugate() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
construction() (sage.rings.rational_field.RationalField method)
content() (sage.rings.rational.Rational method)
continuant() (in module sage.arith.misc)
continued_fraction() (sage.rings.rational.Rational method)
continued_fraction_list() (sage.rings.rational.Rational method)
coprime_integers() (sage.rings.integer.Integer method)
CRT() (in module sage.arith.misc)
crt() (in module sage.arith.misc)
(sage.arith.multi_modular.MultiModularBasis_base method)
(sage.rings.integer.Integer method)
CRT_basis() (in module sage.arith.misc)
crt_basis() (in module sage.rings.integer_ring)
CRT_list() (in module sage.arith.misc)
CRT_vectors() (in module sage.arith.misc)
D
dedekind_psi() (in module sage.arith.misc)
dedekind_sum() (in module sage.arith.misc)
degree() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
denom() (sage.rings.rational.Rational method)
denominator() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
differences() (in module sage.arith.misc)
digits() (sage.rings.integer.Integer method)
discriminant() (sage.rings.rational_field.RationalField method)
divide_knowing_divisible_by() (sage.rings.integer.Integer method)
divides() (sage.rings.integer.Integer method)
divisors() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
E
embeddings() (sage.rings.rational_field.RationalField method)
eratosthenes() (in module sage.arith.misc)
euclidean_degree() (sage.rings.integer.Integer method)
Euler_Phi (class in sage.arith.misc)
exact_log() (sage.rings.integer.Integer method)
exp() (sage.rings.integer.Integer method)
extend_with_primes() (sage.arith.multi_modular.MultiModularBasis_base method)
extension() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
F
factor() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
factor_aurifeuillian() (in module sage.rings.factorint)
factor_cunningham() (in module sage.rings.factorint)
factor_trial_division() (in module sage.rings.factorint)
factor_using_pari() (in module sage.rings.factorint)
factorial() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
falling_factorial() (in module sage.arith.misc)
floor() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
four_squares() (in module sage.arith.misc)
four_squares_pyx() (in module sage.rings.sum_of_squares)
frac() (in module sage.rings.rational_field)
fraction_field() (sage.rings.integer_ring.IntegerRing_class method)
free_integer_pool() (in module sage.rings.integer)
fundamental_discriminant() (in module sage.arith.misc)
G
gamma() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
gauss_sum() (in module sage.arith.misc)
GCD() (in module sage.arith.misc)
gcd() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
gcd_int() (sage.rings.fast_arith.arith_int method)
GCD_list() (in module sage.rings.integer)
gcd_longlong() (sage.rings.fast_arith.arith_llong method)
gen() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
generic_power() (in module sage.arith.power)
gens() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
get_gcd() (in module sage.arith.misc)
get_inverse_mod() (in module sage.arith.misc)
global_height() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
global_height_arch() (sage.rings.rational.Rational method)
global_height_non_arch() (sage.rings.rational.Rational method)
H
height() (sage.rings.rational.Rational method)
hex() (sage.rings.integer.Integer method)
hilbert_conductor() (in module sage.arith.misc)
hilbert_conductor_inverse() (in module sage.arith.misc)
hilbert_symbol() (in module sage.arith.misc)
hilbert_symbol_negative_at_S() (sage.rings.rational_field.RationalField method)
I
imag() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
int_to_Q (class in sage.rings.rational)
int_to_Z (class in sage.rings.integer)
Integer (class in sage.rings.integer)
integer_ceil() (in module sage.arith.misc)
integer_floor() (in module sage.arith.misc)
integer_rational_power() (in module sage.rings.rational)
IntegerRing() (in module sage.rings.integer_ring)
IntegerRing_class (class in sage.rings.integer_ring)
IntegerWrapper (class in sage.rings.integer)
inverse_mod() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
inverse_mod_int() (sage.rings.fast_arith.arith_int method)
inverse_mod_longlong() (sage.rings.fast_arith.arith_llong method)
inverse_of_unit() (sage.rings.integer.Integer method)
is_absolute() (sage.rings.rational_field.RationalField method)
is_field() (sage.rings.integer_ring.IntegerRing_class method)
is_Integer() (in module sage.rings.integer)
is_integer() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_IntegerRing() (in module sage.rings.integer_ring)
is_integral() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_integrally_closed() (sage.rings.integer_ring.IntegerRing_class method)
is_irreducible() (sage.rings.integer.Integer method)
is_noetherian() (sage.rings.integer_ring.IntegerRing_class method)
is_norm() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_nth_power() (sage.rings.rational.Rational method)
is_one() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_padic_square() (sage.rings.rational.Rational method)
is_perfect_power() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_power_of() (sage.rings.integer.Integer method)
is_power_of_two() (in module sage.arith.misc)
is_prime() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
is_prime_field() (sage.rings.rational_field.RationalField method)
is_prime_power() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
is_pseudoprime() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
is_pseudoprime_power() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
is_Rational() (in module sage.rings.rational)
is_rational() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_RationalField() (in module sage.rings.rational_field)
is_S_integral() (sage.rings.rational.Rational method)
is_S_unit() (sage.rings.rational.Rational method)
is_square() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
is_squarefree() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
is_sum_of_two_squares_pyx() (in module sage.rings.sum_of_squares)
is_surjective() (sage.rings.rational.Z_to_Q method)
is_unit() (sage.rings.integer.Integer method)
isqrt() (sage.rings.integer.Integer method)
J
jacobi() (sage.rings.integer.Integer method)
jacobi_symbol() (in module sage.arith.misc)
K
kronecker() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
kronecker_symbol() (in module sage.arith.misc)
krull_dimension() (sage.rings.integer_ring.IntegerRing_class method)
L
lcm() (in module sage.arith.functions)
LCM_list() (in module sage.arith.functions)
legendre_symbol() (in module sage.arith.misc)
list() (sage.arith.multi_modular.MultiModularBasis_base method)
(sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
local_height() (sage.rings.rational.Rational method)
local_height_arch() (sage.rings.rational.Rational method)
log() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
long_to_Q (class in sage.rings.rational)
long_to_Z (class in sage.rings.integer)
M
make_integer() (in module sage.rings.integer)
make_rational() (in module sage.rings.rational)
maximal_order() (sage.rings.rational_field.RationalField method)
minpoly() (sage.rings.rational.Rational method)
mod_ui() (sage.rings.rational.Rational method)
Moebius (class in sage.arith.misc)
mqrr_rational_reconstruction() (in module sage.arith.misc)
multifactorial() (sage.rings.integer.Integer method)
MultiModularBasis (class in sage.arith.multi_modular)
MultiModularBasis_base (class in sage.arith.multi_modular)
multinomial() (in module sage.arith.misc)
multinomial_coefficients() (in module sage.arith.misc)
multiplicative_order() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
MutableMultiModularBasis (class in sage.arith.multi_modular)
N
nbits() (sage.rings.integer.Integer method)
ndigits() (sage.rings.integer.Integer method)
next_prime() (in module sage.arith.misc)
(sage.arith.multi_modular.MutableMultiModularBasis method)
(sage.rings.integer.Integer method)
next_prime_power() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
next_probable_prime() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
ngens() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
norm() (sage.rings.rational.Rational method)
nth_prime() (in module sage.arith.misc)
nth_root() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
number_field() (sage.rings.rational_field.RationalField method)
number_of_divisors() (in module sage.arith.misc)
numer() (sage.rings.rational.Rational method)
numerator() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
O
oct() (sage.rings.integer.Integer method)
odd_part() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
ord() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
order() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
ordinal_str() (sage.rings.integer.Integer method)
P
parameter() (sage.rings.integer_ring.IntegerRing_class method)
partial_product() (sage.arith.multi_modular.MultiModularBasis_base method)
perfect_power() (sage.rings.integer.Integer method)
period() (sage.rings.rational.Rational method)
places() (sage.rings.rational_field.RationalField method)
plot() (sage.arith.misc.Euler_Phi method)
(sage.arith.misc.Moebius method)
(sage.arith.misc.Sigma method)
popcount() (sage.rings.integer.Integer method)
power_basis() (sage.rings.rational_field.RationalField method)
power_mod() (in module sage.arith.misc)
powermod() (sage.rings.integer.Integer method)
powermodm_ui() (sage.rings.integer.Integer method)
precomputation_list() (sage.arith.multi_modular.MultiModularBasis_base method)
previous_prime() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
previous_prime_power() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
prime_divisors() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
prime_factors() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
prime_powers() (in module sage.arith.misc)
prime_range() (in module sage.rings.fast_arith)
prime_to_m_part() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
prime_to_S_part() (sage.rings.rational.Rational method)
primes() (in module sage.arith.misc)
primes_first_n() (in module sage.arith.misc)
primes_of_bounded_norm_iter() (sage.rings.rational_field.RationalField method)
primitive_root() (in module sage.arith.misc)
prod() (sage.arith.multi_modular.MultiModularBasis_base method)
Python Enhancement Proposals
PEP 3127
Q
Q (in module sage.rings.rational_field)
Q_to_Z (class in sage.rings.rational)
QQ (in module sage.rings.rational_field)
quadratic_defect() (sage.rings.rational_field.RationalField method)
quadratic_residues() (in module sage.arith.misc)
quo_rem() (sage.rings.integer.Integer method)
quotient() (sage.rings.integer_ring.IntegerRing_class method)
R
radical() (in module sage.arith.misc)
random_element() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
random_prime() (in module sage.arith.misc)
range() (sage.arith.misc.Moebius method)
(sage.rings.integer_ring.IntegerRing_class method)
range_by_height() (sage.rings.rational_field.RationalField method)
Rational (class in sage.rings.rational)
rational_power_parts() (in module sage.rings.rational)
rational_recon_int() (sage.rings.fast_arith.arith_int method)
rational_recon_longlong() (sage.rings.fast_arith.arith_llong method)
rational_reconstruction() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
RationalField (class in sage.rings.rational_field)
real() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
relative_discriminant() (sage.rings.rational_field.RationalField method)
relative_norm() (sage.rings.rational.Rational method)
replace_prime() (sage.arith.multi_modular.MutableMultiModularBasis method)
residue_field() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
rising_factorial() (in module sage.arith.misc)
round() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
S
sage.arith.functions (module)
sage.arith.misc (module)
sage.arith.multi_modular (module)
sage.arith.power (module)
sage.rings.bernmm (module)
sage.rings.bernoulli_mod_p (module)
sage.rings.factorint (module)
sage.rings.fast_arith (module)
sage.rings.integer (module)
sage.rings.integer_ring (module)
sage.rings.rational (module)
sage.rings.rational_field (module)
sage.rings.sum_of_squares (module)
section() (sage.rings.rational.Q_to_Z method)
(sage.rings.rational.Z_to_Q method)
selmer_group() (sage.rings.rational_field.RationalField method)
selmer_group_iterator() (sage.rings.rational_field.RationalField method)
Sigma (class in sage.arith.misc)
sign() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
signature() (sage.rings.rational_field.RationalField method)
some_elements() (sage.rings.rational_field.RationalField method)
sort_complex_numbers_for_display() (in module sage.arith.misc)
sqrt() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
sqrtrem() (sage.rings.integer.Integer method)
squarefree_divisors() (in module sage.arith.misc)
squarefree_part() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
str() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
subfactorial() (in module sage.arith.misc)
sum_of_k_squares() (in module sage.arith.misc)
support() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
T
test_bit() (sage.rings.integer.Integer method)
three_squares() (in module sage.arith.misc)
three_squares_pyx() (in module sage.rings.sum_of_squares)
trace() (sage.rings.rational.Rational method)
trailing_zero_bits() (sage.rings.integer.Integer method)
trial_division() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
trunc() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
two_squares() (in module sage.arith.misc)
two_squares_pyx() (in module sage.rings.sum_of_squares)
V
val_unit() (sage.rings.integer.Integer method)
(sage.rings.rational.Rational method)
valuation() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
(sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational.Rational method)
(sage.rings.rational_field.RationalField method)
verify_bernoulli_mod_p() (in module sage.rings.bernoulli_mod_p)
X
XGCD() (in module sage.arith.misc)
xgcd() (in module sage.arith.misc)
(sage.rings.integer.Integer method)
xgcd_int() (sage.rings.fast_arith.arith_int method)
xkcd() (in module sage.arith.misc)
xlcm() (in module sage.arith.misc)
Z
Z_to_Q (class in sage.rings.rational)
zeta() (sage.rings.integer_ring.IntegerRing_class method)
(sage.rings.rational_field.RationalField method)
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