def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
if self.is_relaxed():
from sage.rings.padics.pow_computer_flint import PowComputer_flint
self.prime_pow = PowComputer_flint(p, 1, 1, 1, self.is_field())
else:
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec,
self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names,
element_class)
if self.is_field():
if self.is_capped_relative():
coerce_list = [
pAdicCoercion_ZZ_CR(self),
pAdicCoercion_QQ_CR(self)
]
convert_list = []
elif self.is_floating_point():
coerce_list = [
pAdicCoercion_ZZ_FP(self),
pAdicCoercion_QQ_FP(self)
]
convert_list = []
elif self.is_lattice_prec():
coerce_list = [QQ]
convert_list = []
elif self.is_relaxed():
coerce_list = [QQ]
convert_list = []
else:
raise RuntimeError
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self)]
convert_list = [pAdicConvert_QQ_FP(self)]
elif self.is_lattice_prec():
coerce_list = [ZZ]
convert_list = [QQ]
elif self.is_relaxed():
coerce_list = [ZZ]
convert_list = [QQ]
else:
raise RuntimeError
self.Element = element_class
self._populate_coercion_lists_(coerce_list=coerce_list,
convert_list=convert_list)
def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec,
self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names,
element_class)
if self.is_field():
coerce_list = [
pAdicCoercion_ZZ_CR(self),
pAdicCoercion_QQ_CR(self)
]
convert_list = []
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
else:
raise RuntimeError
self._populate_coercion_lists_(coerce_list=coerce_list,
convert_list=convert_list,
element_constructor=element_class)
def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec, self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names, element_class)
if self.is_field():
if self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self), pAdicCoercion_QQ_CR(self)]
convert_list = []
else:
coerce_list = [pAdicCoercion_ZZ_FP(self), pAdicCoercion_QQ_FP(self)]
convert_list = []
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self)]
convert_list = [pAdicConvert_QQ_FP(self)]
else:
raise RuntimeError
self._populate_coercion_lists_(coerce_list=coerce_list, convert_list=convert_list, element_constructor=element_class)
class pAdicBaseGeneric(pAdicGeneric):
_implementation = 'GMP'
def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec, self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names, element_class)
if self.is_field():
if self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self), pAdicCoercion_QQ_CR(self)]
convert_list = []
else:
coerce_list = [pAdicCoercion_ZZ_FP(self), pAdicCoercion_QQ_FP(self)]
convert_list = []
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self)]
convert_list = [pAdicConvert_QQ_FP(self)]
else:
raise RuntimeError
self._populate_coercion_lists_(coerce_list=coerce_list, convert_list=convert_list, element_constructor=element_class)
def fraction_field(self, print_mode=None):
r"""
Returns the fraction field of ``self``.
INPUT:
- ``print_mode`` - a dictionary containing print options.
Defaults to the same options as this ring.
OUTPUT:
- the fraction field of ``self``.
EXAMPLES::
sage: R = Zp(5, print_mode='digits')
sage: K = R.fraction_field(); repr(K(1/3))[3:]
'31313131313131313132'
sage: L = R.fraction_field({'max_ram_terms':4}); repr(L(1/3))[3:]
'3132'
"""
if self.is_field() and print_mode is None:
return self
from sage.rings.padics.factory import Qp
if self.is_floating_point():
mode = 'floating-point'
else:
mode = 'capped-rel'
return Qp(self.prime(), self.precision_cap(), mode, print_mode=self._modified_print_mode(print_mode), names=self._uniformizer_print())
def integer_ring(self, print_mode=None):
r"""
Returns the integer ring of ``self``, possibly with
``print_mode`` changed.
INPUT:
- ``print_mode`` - a dictionary containing print options.
Defaults to the same options as this ring.
OUTPUT:
- The ring of integral elements in ``self``.
EXAMPLES::
sage: K = Qp(5, print_mode='digits')
sage: R = K.integer_ring(); repr(R(1/3))[3:]
'31313131313131313132'
sage: S = K.integer_ring({'max_ram_terms':4}); repr(S(1/3))[3:]
'3132'
"""
if not self.is_field() and print_mode is None:
return self
from sage.rings.padics.factory import Zp
return Zp(self.prime(), self.precision_cap(), self._prec_type(), print_mode=self._modified_print_mode(print_mode), names=self._uniformizer_print())
def is_isomorphic(self, ring):
r"""
Returns whether ``self`` and ``ring`` are isomorphic, i.e. whether ``ring`` is an implementation of `\mathbb{Z}_p` for the same prime as ``self``.
INPUT:
- ``self`` -- a `p`-adic ring
- ``ring`` -- a ring
OUTPUT:
- ``boolean`` -- whether ``ring`` is an implementation of \mathbb{Z}_p` for the same prime as ``self``.
EXAMPLES::
sage: R = Zp(5, 15, print_mode='digits'); S = Zp(5, 44, print_max_terms=4); R.is_isomorphic(S)
True
"""
return isinstance(ring, pAdicBaseGeneric) and self.prime() == ring.prime() and self.is_field() == ring.is_field()
def gen(self, n=0):
"""
Returns the ``nth`` generator of this extension. For base
rings/fields, we consider the generator to be the prime.
EXAMPLES::
sage: R = Zp(5); R.gen()
5 + O(5^21)
"""
if n != 0:
raise IndexError("only one generator")
return self(self.prime())
def absolute_discriminant(self):
"""
Returns the absolute discriminant of this `p`-adic ring
EXAMPLES::
sage: Zp(5).absolute_discriminant()
1
"""
return 1
def discriminant(self, K=None):
"""
Returns the discriminant of this `p`-adic ring over ``K``
INPUT:
- ``self`` -- a `p`-adic ring
- ``K`` -- a sub-ring of ``self`` or ``None`` (default: ``None``)
OUTPUT:
- integer -- the discriminant of this ring over ``K`` (or the
absolute discriminant if ``K`` is ``None``)
EXAMPLES::
sage: Zp(5).discriminant()
1
"""
if (K is None or K is self):
return 1
else:
raise ValueError("Ground Ring must be a subring of self")
def is_abelian(self):
"""
Returns whether the Galois group is abelian, i.e. ``True``.
#should this be automorphism group?
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_abelian()
True
"""
return True
def is_normal(self):
"""
Returns whether or not this is a normal extension, i.e. ``True``.
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_normal()
True
"""
return True
def uniformizer(self):
"""
Returns a uniformizer for this ring.
EXAMPLES::
sage: R = Zp(3,5,'fixed-mod', 'series')
sage: R.uniformizer()
3 + O(3^5)
"""
return self(self.prime_pow._prime())
def uniformizer_pow(self, n):
"""
Returns the ``nth`` power of the uniformizer of ``self`` (as
an element of ``self``).
EXAMPLES::
sage: R = Zp(5)
sage: R.uniformizer_pow(5)
5^5 + O(5^25)
sage: R.uniformizer_pow(infinity)
0
"""
return self(self.prime_pow(n))
def _uniformizer_print(self):
"""
Returns how the uniformizer is supposed to print.
EXAMPLES::
sage: R = Zp(5, names='pi'); R._uniformizer_print()
'pi'
"""
return self.variable_name()
def has_pth_root(self):
r"""
Returns whether or not `\mathbb{Z}_p` has a primitive `p^{th}`
root of unity.
EXAMPLES::
sage: Zp(2).has_pth_root()
True
sage: Zp(17).has_pth_root()
False
"""
return (self.prime() == 2)
def has_root_of_unity(self, n):
r"""
Returns whether or not `\mathbb{Z}_p` has a primitive `n^{th}`
root of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer
OUTPUT:
- ``boolean`` -- whether ``self`` has primitive `n^{th}` root
of unity
EXAMPLES::
sage: R=Zp(37)
sage: R.has_root_of_unity(12)
True
sage: R.has_root_of_unity(11)
False
"""
if (self.prime() == 2):
return n.divides(2)
else:
return n.divides(self.prime() - 1)
def zeta(self, n=None):
r"""
Returns a generator of the group of roots of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer or ``None`` (default: ``None``)
OUTPUT:
- ``element`` -- a generator of the `n^{th}` roots of unity,
or a generator of the full group of roots of unity if ``n``
is ``None``
EXAMPLES::
sage: R = Zp(37,5)
sage: R.zeta(12)
8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
"""
if (self.prime() == 2):
if (n is None) or (n == 2):
return self(-1)
if n == 1:
return self(1)
else:
raise ValueError("No, %sth root of unity in self"%n)
else:
from sage.rings.finite_rings.finite_field_constructor import GF
return self.teichmuller(GF(self.prime()).zeta(n).lift())
def zeta_order(self):
"""
Returns the order of the group of roots of unity.
EXAMPLES::
sage: R = Zp(37); R.zeta_order()
36
sage: Zp(2).zeta_order()
2
"""
if (self.prime() == 2):
return 2
else:
return self.prime() - 1
def plot(self, max_points=2500, **args):
r"""
Create a visualization of this `p`-adic ring as a fractal
similar to a generalization of the Sierpi\'nski
triangle.
The resulting image attempts to capture the
algebraic and topological characteristics of `\mathbb{Z}_p`.
INPUT:
- ``max_points`` -- the maximum number or points to plot,
which controls the depth of recursion (default 2500)
- ``**args`` -- color, size, etc. that are passed to the
underlying point graphics objects
REFERENCES:
- Cuoco, A. ''Visualizing the `p`-adic Integers'', The
American Mathematical Monthly, Vol. 98, No. 4 (Apr., 1991),
pp. 355-364
EXAMPLES::
sage: Zp(3).plot()
Graphics object consisting of 1 graphics primitive
sage: Zp(5).plot(max_points=625)
Graphics object consisting of 1 graphics primitive
sage: Zp(23).plot(rgbcolor=(1,0,0))
Graphics object consisting of 1 graphics primitive
"""
if 'pointsize' not in args:
args['pointsize'] = 1
from sage.misc.mrange import cartesian_product_iterator
from sage.rings.real_double import RDF
from sage.plot.all import points, circle, Graphics
p = self.prime()
phi = 2*RDF.pi()/p
V = RDF**2
vs = [V([(phi*t).sin(), (phi*t).cos()]) for t in range(p)]
all = []
depth = max(RDF(max_points).log(p).floor(), 1)
scale = min(RDF(1.5/p), 1/RDF(3))
pts = [vs]*depth
if depth == 1 and 23 < p < max_points:
extras = int(max_points/p)
if p/extras > 5:
pts = [vs]*depth + [vs[::extras]]
for digits in cartesian_product_iterator(pts):
p = sum([v * scale**n for n, v in enumerate(digits)])
all.append(tuple(p))
g = points(all, **args)
# Set default plotting options
g.axes(False)
g.set_aspect_ratio(1)
return g
class pAdicBaseGeneric(pAdicGeneric):
_implementation = 'GMP'
def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec,
self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names,
element_class)
if self.is_field():
if self.is_capped_relative():
coerce_list = [
pAdicCoercion_ZZ_CR(self),
pAdicCoercion_QQ_CR(self)
]
convert_list = []
else:
coerce_list = [
pAdicCoercion_ZZ_FP(self),
pAdicCoercion_QQ_FP(self)
]
convert_list = []
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self)]
convert_list = [pAdicConvert_QQ_FP(self)]
else:
raise RuntimeError
self._populate_coercion_lists_(coerce_list=coerce_list,
convert_list=convert_list,
element_constructor=element_class)
def fraction_field(self, print_mode=None):
r"""
Returns the fraction field of ``self``.
INPUT:
- ``print_mode`` - a dictionary containing print options.
Defaults to the same options as this ring.
OUTPUT:
- the fraction field of ``self``.
EXAMPLES::
sage: R = Zp(5, print_mode='digits')
sage: K = R.fraction_field(); repr(K(1/3))[3:]
'31313131313131313132'
sage: L = R.fraction_field({'max_ram_terms':4}); repr(L(1/3))[3:]
'3132'
"""
if self.is_field() and print_mode is None:
return self
from sage.rings.padics.factory import Qp
if self.is_floating_point():
mode = 'floating-point'
else:
mode = 'capped-rel'
return Qp(self.prime(),
self.precision_cap(),
mode,
print_mode=self._modified_print_mode(print_mode),
names=self._uniformizer_print())
def integer_ring(self, print_mode=None):
r"""
Returns the integer ring of ``self``, possibly with
``print_mode`` changed.
INPUT:
- ``print_mode`` - a dictionary containing print options.
Defaults to the same options as this ring.
OUTPUT:
- The ring of integral elements in ``self``.
EXAMPLES::
sage: K = Qp(5, print_mode='digits')
sage: R = K.integer_ring(); repr(R(1/3))[3:]
'31313131313131313132'
sage: S = K.integer_ring({'max_ram_terms':4}); repr(S(1/3))[3:]
'3132'
"""
if not self.is_field() and print_mode is None:
return self
from sage.rings.padics.factory import Zp
return Zp(self.prime(),
self.precision_cap(),
self._prec_type(),
print_mode=self._modified_print_mode(print_mode),
names=self._uniformizer_print())
def is_isomorphic(self, ring):
r"""
Returns whether ``self`` and ``ring`` are isomorphic, i.e. whether ``ring`` is an implementation of `\mathbb{Z}_p` for the same prime as ``self``.
INPUT:
- ``self`` -- a `p`-adic ring
- ``ring`` -- a ring
OUTPUT:
- ``boolean`` -- whether ``ring`` is an implementation of \mathbb{Z}_p` for the same prime as ``self``.
EXAMPLES::
sage: R = Zp(5, 15, print_mode='digits'); S = Zp(5, 44, print_max_terms=4); R.is_isomorphic(S)
True
"""
return isinstance(ring, pAdicBaseGeneric) and self.prime(
) == ring.prime() and self.is_field() == ring.is_field()
def gen(self, n=0):
"""
Returns the ``nth`` generator of this extension. For base
rings/fields, we consider the generator to be the prime.
EXAMPLES::
sage: R = Zp(5); R.gen()
5 + O(5^21)
"""
if n != 0:
raise IndexError("only one generator")
return self(self.prime())
def absolute_discriminant(self):
"""
Returns the absolute discriminant of this `p`-adic ring
EXAMPLES::
sage: Zp(5).absolute_discriminant()
1
"""
return 1
def discriminant(self, K=None):
"""
Returns the discriminant of this `p`-adic ring over ``K``
INPUT:
- ``self`` -- a `p`-adic ring
- ``K`` -- a sub-ring of ``self`` or ``None`` (default: ``None``)
OUTPUT:
- integer -- the discriminant of this ring over ``K`` (or the
absolute discriminant if ``K`` is ``None``)
EXAMPLES::
sage: Zp(5).discriminant()
1
"""
if (K is None or K is self):
return 1
else:
raise ValueError("Ground Ring must be a subring of self")
def is_abelian(self):
"""
Returns whether the Galois group is abelian, i.e. ``True``.
#should this be automorphism group?
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_abelian()
True
"""
return True
def is_normal(self):
"""
Returns whether or not this is a normal extension, i.e. ``True``.
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_normal()
True
"""
return True
def uniformizer(self):
"""
Returns a uniformizer for this ring.
EXAMPLES::
sage: R = Zp(3,5,'fixed-mod', 'series')
sage: R.uniformizer()
3 + O(3^5)
"""
return self(self.prime_pow._prime())
def uniformizer_pow(self, n):
"""
Returns the ``nth`` power of the uniformizer of ``self`` (as
an element of ``self``).
EXAMPLES::
sage: R = Zp(5)
sage: R.uniformizer_pow(5)
5^5 + O(5^25)
sage: R.uniformizer_pow(infinity)
0
"""
return self(self.prime_pow(n))
def _uniformizer_print(self):
"""
Returns how the uniformizer is supposed to print.
EXAMPLES::
sage: R = Zp(5, names='pi'); R._uniformizer_print()
'pi'
"""
return self.variable_name()
def has_pth_root(self):
r"""
Returns whether or not `\mathbb{Z}_p` has a primitive `p^{th}`
root of unity.
EXAMPLES::
sage: Zp(2).has_pth_root()
True
sage: Zp(17).has_pth_root()
False
"""
return (self.prime() == 2)
def has_root_of_unity(self, n):
r"""
Returns whether or not `\mathbb{Z}_p` has a primitive `n^{th}`
root of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer
OUTPUT:
- ``boolean`` -- whether ``self`` has primitive `n^{th}` root
of unity
EXAMPLES::
sage: R=Zp(37)
sage: R.has_root_of_unity(12)
True
sage: R.has_root_of_unity(11)
False
"""
if (self.prime() == 2):
return n.divides(2)
else:
return n.divides(self.prime() - 1)
def zeta(self, n=None):
r"""
Returns a generator of the group of roots of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer or ``None`` (default: ``None``)
OUTPUT:
- ``element`` -- a generator of the `n^{th}` roots of unity,
or a generator of the full group of roots of unity if ``n``
is ``None``
EXAMPLES::
sage: R = Zp(37,5)
sage: R.zeta(12)
8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
"""
if (self.prime() == 2):
if (n is None) or (n == 2):
return self(-1)
if n == 1:
return self(1)
else:
raise ValueError("No, %sth root of unity in self" % n)
else:
from sage.rings.finite_rings.finite_field_constructor import GF
return self.teichmuller(GF(self.prime()).zeta(n).lift())
def zeta_order(self):
"""
Returns the order of the group of roots of unity.
EXAMPLES::
sage: R = Zp(37); R.zeta_order()
36
sage: Zp(2).zeta_order()
2
"""
if (self.prime() == 2):
return 2
else:
return self.prime() - 1
def plot(self, max_points=2500, **args):
r"""
Create a visualization of this `p`-adic ring as a fractal
similar to a generalization of the Sierpi\'nski
triangle.
The resulting image attempts to capture the
algebraic and topological characteristics of `\mathbb{Z}_p`.
INPUT:
- ``max_points`` -- the maximum number or points to plot,
which controls the depth of recursion (default 2500)
- ``**args`` -- color, size, etc. that are passed to the
underlying point graphics objects
REFERENCES:
- Cuoco, A. ''Visualizing the `p`-adic Integers'', The
American Mathematical Monthly, Vol. 98, No. 4 (Apr., 1991),
pp. 355-364
EXAMPLES::
sage: Zp(3).plot()
Graphics object consisting of 1 graphics primitive
sage: Zp(5).plot(max_points=625)
Graphics object consisting of 1 graphics primitive
sage: Zp(23).plot(rgbcolor=(1,0,0))
Graphics object consisting of 1 graphics primitive
"""
if 'pointsize' not in args:
args['pointsize'] = 1
from sage.misc.mrange import cartesian_product_iterator
from sage.rings.real_double import RDF
from sage.plot.all import points, circle, Graphics
p = self.prime()
phi = 2 * RDF.pi() / p
V = RDF**2
vs = [V([(phi * t).sin(), (phi * t).cos()]) for t in range(p)]
all = []
depth = max(RDF(max_points).log(p).floor(), 1)
scale = min(RDF(1.5 / p), 1 / RDF(3))
pts = [vs] * depth
if depth == 1 and 23 < p < max_points:
extras = int(max_points / p)
if p / extras > 5:
pts = [vs] * depth + [vs[::extras]]
for digits in cartesian_product_iterator(pts):
p = sum([v * scale**n for n, v in enumerate(digits)])
all.append(tuple(p))
g = points(all, **args)
# Set default plotting options
g.axes(False)
g.set_aspect_ratio(1)
return g
class pAdicBaseGeneric(pAdicGeneric):
_implementation = 'GMP'
def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
if self.is_relaxed():
from sage.rings.padics.pow_computer_flint import PowComputer_flint
self.prime_pow = PowComputer_flint(p, 1, 1, 1, self.is_field())
else:
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec,
self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names,
element_class)
if self.is_field():
if self.is_capped_relative():
coerce_list = [
pAdicCoercion_ZZ_CR(self),
pAdicCoercion_QQ_CR(self)
]
convert_list = []
elif self.is_floating_point():
coerce_list = [
pAdicCoercion_ZZ_FP(self),
pAdicCoercion_QQ_FP(self)
]
convert_list = []
elif self.is_lattice_prec():
coerce_list = [QQ]
convert_list = []
elif self.is_relaxed():
coerce_list = [QQ]
convert_list = []
else:
raise RuntimeError
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self)]
convert_list = [pAdicConvert_QQ_FP(self)]
elif self.is_lattice_prec():
coerce_list = [ZZ]
convert_list = [QQ]
elif self.is_relaxed():
coerce_list = [ZZ]
convert_list = [QQ]
else:
raise RuntimeError
self.Element = element_class
self._populate_coercion_lists_(coerce_list=coerce_list,
convert_list=convert_list)
def _repr_(self, do_latex=False):
r"""
Returns a print representation of this p-adic ring or field.
EXAMPLES::
sage: K = Zp(17); K #indirect doctest
17-adic Ring with capped relative precision 20
sage: latex(K)
\Bold{Z}_{17}
sage: K = ZpCA(17); K #indirect doctest
17-adic Ring with capped absolute precision 20
sage: latex(K)
\Bold{Z}_{17}
sage: K = ZpFP(17); K #indirect doctest
17-adic Ring with floating precision 20
sage: latex(K)
\Bold{Z}_{17}
sage: K = ZpFM(7); K
7-adic Ring of fixed modulus 7^20
sage: latex(K) #indirect doctest
\Bold{Z}_{7}
sage: K = ZpLF(2); K # indirect doctest
doctest:...: FutureWarning: This class/method/function is marked as experimental. It, its functionality or its interface might change without a formal deprecation.
See http://trac.sagemath.org/23505 for details.
2-adic Ring with lattice-float precision
sage: latex(K)
\Bold{Z}_{2}
sage: K = Qp(17); K #indirect doctest
17-adic Field with capped relative precision 20
sage: latex(K)
\Bold{Q}_{17}
sage: K = QpFP(17); K #indirect doctest
17-adic Field with floating precision 20
sage: latex(K)
\Bold{Q}_{17}
sage: K = QpLC(2); K # indirect doctest
2-adic Field with lattice-cap precision
sage: latex(K)
\Bold{Q}_{2}
"""
if do_latex:
if self.is_field():
s = r"\Bold{Q}_{%s}" % self.prime()
else:
s = r"\Bold{Z}_{%s}" % self.prime()
if hasattr(self, '_label') and self._label:
s = r"\verb'%s' (\simeq %s)" % (self._label, s)
else:
s = "Field " if self.is_field() else "Ring "
s = "%s-adic " % self.prime() + s + precprint(
self._prec_type(), self.precision_cap(), self.prime())
if hasattr(self, '_label') and self._label:
s += " (label: %s)" % self._label
return s
def exact_field(self):
"""
Returns the rational field.
For compatibility with extensions of p-adics.
EXAMPLES::
sage: Zp(5).exact_field()
Rational Field
"""
from sage.rings.rational_field import QQ
return QQ
def exact_ring(self):
"""
Returns the integer ring.
EXAMPLES::
sage: Zp(5).exact_ring()
Integer Ring
"""
from sage.rings.integer_ring import ZZ
return ZZ
def is_isomorphic(self, ring):
r"""
Returns whether ``self`` and ``ring`` are isomorphic, i.e. whether ``ring`` is an implementation of `\ZZ_p` for the same prime as ``self``.
INPUT:
- ``self`` -- a `p`-adic ring
- ``ring`` -- a ring
OUTPUT:
- ``boolean`` -- whether ``ring`` is an implementation of \ZZ_p` for the same prime as ``self``.
EXAMPLES::
sage: R = Zp(5, 15, print_mode='digits'); S = Zp(5, 44, print_max_terms=4); R.is_isomorphic(S)
True
"""
return isinstance(ring, pAdicBaseGeneric) and self.prime(
) == ring.prime() and self.is_field() == ring.is_field()
def gen(self, n=0):
"""
Returns the ``nth`` generator of this extension. For base
rings/fields, we consider the generator to be the prime.
EXAMPLES::
sage: R = Zp(5); R.gen()
5 + O(5^21)
"""
if n != 0:
raise IndexError("only one generator")
return self(self.prime())
def modulus(self, exact=False):
r"""
Returns the polynomial defining this extension.
For compatibility with extension fields; we define the modulus to be x-1.
INPUT:
- ``exact`` -- boolean (default ``False``), whether to return a polynomial with integer entries.
EXAMPLES::
sage: Zp(5).modulus(exact=True)
x
"""
return self.defining_polynomial(exact=exact)
def absolute_discriminant(self):
"""
Returns the absolute discriminant of this `p`-adic ring
EXAMPLES::
sage: Zp(5).absolute_discriminant()
1
"""
return 1
def discriminant(self, K=None):
"""
Returns the discriminant of this `p`-adic ring over ``K``
INPUT:
- ``self`` -- a `p`-adic ring
- ``K`` -- a sub-ring of ``self`` or ``None`` (default: ``None``)
OUTPUT:
- integer -- the discriminant of this ring over ``K`` (or the
absolute discriminant if ``K`` is ``None``)
EXAMPLES::
sage: Zp(5).discriminant()
1
"""
if (K is None or K is self):
return 1
else:
raise ValueError("Ground Ring must be a subring of self")
def is_abelian(self):
"""
Returns whether the Galois group is abelian, i.e. ``True``.
#should this be automorphism group?
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_abelian()
True
"""
return True
def is_normal(self):
"""
Returns whether or not this is a normal extension, i.e. ``True``.
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_normal()
True
"""
return True
def uniformizer(self):
"""
Returns a uniformizer for this ring.
EXAMPLES::
sage: R = Zp(3,5,'fixed-mod', 'series')
sage: R.uniformizer()
3
"""
return self(self.prime_pow._prime())
def uniformizer_pow(self, n):
"""
Returns the ``nth`` power of the uniformizer of ``self`` (as
an element of ``self``).
EXAMPLES::
sage: R = Zp(5)
sage: R.uniformizer_pow(5)
5^5 + O(5^25)
sage: R.uniformizer_pow(infinity)
0
"""
return self(self.prime_pow(n))
def _uniformizer_print(self):
"""
Returns how the uniformizer is supposed to print.
EXAMPLES::
sage: R = Zp(5, names='pi'); R._uniformizer_print()
'pi'
"""
return self.variable_name()
def has_pth_root(self):
r"""
Returns whether or not `\ZZ_p` has a primitive `p^{th}`
root of unity.
EXAMPLES::
sage: Zp(2).has_pth_root()
True
sage: Zp(17).has_pth_root()
False
"""
return (self.prime() == 2)
def has_root_of_unity(self, n):
r"""
Returns whether or not `\ZZ_p` has a primitive `n^{th}`
root of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer
OUTPUT:
- ``boolean`` -- whether ``self`` has primitive `n^{th}` root
of unity
EXAMPLES::
sage: R=Zp(37)
sage: R.has_root_of_unity(12)
True
sage: R.has_root_of_unity(11)
False
"""
if (self.prime() == 2):
return n.divides(2)
else:
return n.divides(self.prime() - 1)
def zeta(self, n=None):
r"""
Returns a generator of the group of roots of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer or ``None`` (default: ``None``)
OUTPUT:
- ``element`` -- a generator of the `n^{th}` roots of unity,
or a generator of the full group of roots of unity if ``n``
is ``None``
EXAMPLES::
sage: R = Zp(37,5)
sage: R.zeta(12)
8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
"""
if (self.prime() == 2):
if (n is None) or (n == 2):
return self(-1)
if n == 1:
return self(1)
else:
raise ValueError("No, %sth root of unity in self" % n)
else:
from sage.rings.finite_rings.finite_field_constructor import GF
return self.teichmuller(GF(self.prime()).zeta(n).lift())
def zeta_order(self):
"""
Returns the order of the group of roots of unity.
EXAMPLES::
sage: R = Zp(37); R.zeta_order()
36
sage: Zp(2).zeta_order()
2
"""
if (self.prime() == 2):
return 2
else:
return self.prime() - 1
def plot(self, max_points=2500, **args):
r"""
Create a visualization of this `p`-adic ring as a fractal
similar to a generalization of the Sierpi\'nski
triangle.
The resulting image attempts to capture the
algebraic and topological characteristics of `\ZZ_p`.
INPUT:
- ``max_points`` -- the maximum number or points to plot,
which controls the depth of recursion (default 2500)
- ``**args`` -- color, size, etc. that are passed to the
underlying point graphics objects
REFERENCES:
- Cuoco, A. ''Visualizing the `p`-adic Integers'', The
American Mathematical Monthly, Vol. 98, No. 4 (Apr., 1991),
pp. 355-364
EXAMPLES::
sage: Zp(3).plot()
Graphics object consisting of 1 graphics primitive
sage: Zp(5).plot(max_points=625)
Graphics object consisting of 1 graphics primitive
sage: Zp(23).plot(rgbcolor=(1,0,0))
Graphics object consisting of 1 graphics primitive
"""
if 'pointsize' not in args:
args['pointsize'] = 1
from sage.misc.mrange import cartesian_product_iterator
from sage.rings.real_double import RDF
from sage.plot.all import points
p = self.prime()
phi = 2 * RDF.pi() / p
V = RDF**2
vs = [V([(phi * t).sin(), (phi * t).cos()]) for t in range(p)]
all = []
depth = max(RDF(max_points).log(p).floor(), 1)
scale = min(RDF(1.5 / p), 1 / RDF(3))
pts = [vs] * depth
if depth == 1 and 23 < p < max_points:
extras = int(max_points / p)
if p / extras > 5:
pts = [vs] * depth + [vs[::extras]]
for digits in cartesian_product_iterator(pts):
p = sum([v * scale**n for n, v in enumerate(digits)])
all.append(tuple(p))
g = points(all, **args)
# Set default plotting options
g.axes(False)
g.set_aspect_ratio(1)
return g
class pAdicBaseGeneric(pAdicGeneric):
_implementation = 'GMP'
def __init__(self, p, prec, print_mode, names, element_class):
"""
Initialization
TESTS::
sage: R = Zp(5) #indirect doctest
"""
self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec, self.is_field(), self._prec_type())
pAdicGeneric.__init__(self, self, p, prec, print_mode, names, element_class)
if self.is_field():
if self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self), pAdicCoercion_QQ_CR(self)]
convert_list = []
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self), pAdicCoercion_QQ_FP(self)]
convert_list = []
elif self.is_lattice_prec():
coerce_list = [QQ]
convert_list = []
else:
raise RuntimeError
elif self.is_capped_relative():
coerce_list = [pAdicCoercion_ZZ_CR(self)]
convert_list = [pAdicConvert_QQ_CR(self)]
elif self.is_capped_absolute():
coerce_list = [pAdicCoercion_ZZ_CA(self)]
convert_list = [pAdicConvert_QQ_CA(self)]
elif self.is_fixed_mod():
coerce_list = [pAdicCoercion_ZZ_FM(self)]
convert_list = [pAdicConvert_QQ_FM(self)]
elif self.is_floating_point():
coerce_list = [pAdicCoercion_ZZ_FP(self)]
convert_list = [pAdicConvert_QQ_FP(self)]
elif self.is_lattice_prec():
coerce_list = [ZZ]
convert_list = [QQ]
else:
raise RuntimeError
self.Element = element_class
self._populate_coercion_lists_(coerce_list=coerce_list, convert_list=convert_list)
def _repr_(self, do_latex=False):
r"""
Returns a print representation of this p-adic ring or field.
EXAMPLES::
sage: K = Zp(17); K #indirect doctest
17-adic Ring with capped relative precision 20
sage: latex(K)
\ZZ_{17}
sage: K = ZpCA(17); K #indirect doctest
17-adic Ring with capped absolute precision 20
sage: latex(K)
\ZZ_{17}
sage: K = ZpFP(17); K #indirect doctest
17-adic Ring with floating precision 20
sage: latex(K)
\ZZ_{17}
sage: K = ZpFM(7); K
7-adic Ring of fixed modulus 7^20
sage: latex(K) #indirect doctest
\ZZ_{7}
sage: K = ZpLF(2); K # indirect doctest
doctest:...: FutureWarning: This class/method/function is marked as experimental. It, its functionality or its interface might change without a formal deprecation.
See http://trac.sagemath.org/23505 for details.
2-adic Ring with lattice-float precision
sage: latex(K)
\ZZ_{2}
sage: K = Qp(17); K #indirect doctest
17-adic Field with capped relative precision 20
sage: latex(K)
\QQ_{17}
sage: K = QpFP(17); K #indirect doctest
17-adic Field with floating precision 20
sage: latex(K)
\QQ_{17}
sage: K = QpLC(2); K # indirect doctest
2-adic Field with lattice-cap precision
sage: latex(K)
\QQ_{2}
"""
if do_latex:
if self.is_field():
s = r"\QQ_{%s}" % self.prime()
else:
s = r"\ZZ_{%s}" % self.prime()
if hasattr(self, '_label') and self._label:
s = r"\verb'%s' (\simeq %s)"%(self._label, s)
else:
s = "Field " if self.is_field() else "Ring "
s = "%s-adic "%self.prime() + s + precprint(self._prec_type(), self.precision_cap(), self.prime())
if hasattr(self, '_label') and self._label:
s+= " (label: %s)"%self._label
return s
def exact_field(self):
"""
Returns the rational field.
For compatibility with extensions of p-adics.
EXAMPLES::
sage: Zp(5).exact_field()
Rational Field
"""
from sage.rings.rational_field import QQ
return QQ
def exact_ring(self):
"""
Returns the integer ring.
EXAMPLES::
sage: Zp(5).exact_ring()
Integer Ring
"""
from sage.rings.integer_ring import ZZ
return ZZ
def is_isomorphic(self, ring):
r"""
Returns whether ``self`` and ``ring`` are isomorphic, i.e. whether ``ring`` is an implementation of `\mathbb{Z}_p` for the same prime as ``self``.
INPUT:
- ``self`` -- a `p`-adic ring
- ``ring`` -- a ring
OUTPUT:
- ``boolean`` -- whether ``ring`` is an implementation of \mathbb{Z}_p` for the same prime as ``self``.
EXAMPLES::
sage: R = Zp(5, 15, print_mode='digits'); S = Zp(5, 44, print_max_terms=4); R.is_isomorphic(S)
True
"""
return isinstance(ring, pAdicBaseGeneric) and self.prime() == ring.prime() and self.is_field() == ring.is_field()
def gen(self, n=0):
"""
Returns the ``nth`` generator of this extension. For base
rings/fields, we consider the generator to be the prime.
EXAMPLES::
sage: R = Zp(5); R.gen()
5 + O(5^21)
"""
if n != 0:
raise IndexError("only one generator")
return self(self.prime())
def absolute_discriminant(self):
"""
Returns the absolute discriminant of this `p`-adic ring
EXAMPLES::
sage: Zp(5).absolute_discriminant()
1
"""
return 1
def discriminant(self, K=None):
"""
Returns the discriminant of this `p`-adic ring over ``K``
INPUT:
- ``self`` -- a `p`-adic ring
- ``K`` -- a sub-ring of ``self`` or ``None`` (default: ``None``)
OUTPUT:
- integer -- the discriminant of this ring over ``K`` (or the
absolute discriminant if ``K`` is ``None``)
EXAMPLES::
sage: Zp(5).discriminant()
1
"""
if (K is None or K is self):
return 1
else:
raise ValueError("Ground Ring must be a subring of self")
def is_abelian(self):
"""
Returns whether the Galois group is abelian, i.e. ``True``.
#should this be automorphism group?
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_abelian()
True
"""
return True
def is_normal(self):
"""
Returns whether or not this is a normal extension, i.e. ``True``.
EXAMPLES::
sage: R = Zp(3, 10,'fixed-mod'); R.is_normal()
True
"""
return True
def uniformizer(self):
"""
Returns a uniformizer for this ring.
EXAMPLES::
sage: R = Zp(3,5,'fixed-mod', 'series')
sage: R.uniformizer()
3 + O(3^5)
"""
return self(self.prime_pow._prime())
def uniformizer_pow(self, n):
"""
Returns the ``nth`` power of the uniformizer of ``self`` (as
an element of ``self``).
EXAMPLES::
sage: R = Zp(5)
sage: R.uniformizer_pow(5)
5^5 + O(5^25)
sage: R.uniformizer_pow(infinity)
0
"""
return self(self.prime_pow(n))
def _uniformizer_print(self):
"""
Returns how the uniformizer is supposed to print.
EXAMPLES::
sage: R = Zp(5, names='pi'); R._uniformizer_print()
'pi'
"""
return self.variable_name()
def has_pth_root(self):
r"""
Returns whether or not `\mathbb{Z}_p` has a primitive `p^{th}`
root of unity.
EXAMPLES::
sage: Zp(2).has_pth_root()
True
sage: Zp(17).has_pth_root()
False
"""
return (self.prime() == 2)
def has_root_of_unity(self, n):
r"""
Returns whether or not `\mathbb{Z}_p` has a primitive `n^{th}`
root of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer
OUTPUT:
- ``boolean`` -- whether ``self`` has primitive `n^{th}` root
of unity
EXAMPLES::
sage: R=Zp(37)
sage: R.has_root_of_unity(12)
True
sage: R.has_root_of_unity(11)
False
"""
if (self.prime() == 2):
return n.divides(2)
else:
return n.divides(self.prime() - 1)
def zeta(self, n=None):
r"""
Returns a generator of the group of roots of unity.
INPUT:
- ``self`` -- a `p`-adic ring
- ``n`` -- an integer or ``None`` (default: ``None``)
OUTPUT:
- ``element`` -- a generator of the `n^{th}` roots of unity,
or a generator of the full group of roots of unity if ``n``
is ``None``
EXAMPLES::
sage: R = Zp(37,5)
sage: R.zeta(12)
8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
"""
if (self.prime() == 2):
if (n is None) or (n == 2):
return self(-1)
if n == 1:
return self(1)
else:
raise ValueError("No, %sth root of unity in self"%n)
else:
from sage.rings.finite_rings.finite_field_constructor import GF
return self.teichmuller(GF(self.prime()).zeta(n).lift())
def zeta_order(self):
"""
Returns the order of the group of roots of unity.
EXAMPLES::
sage: R = Zp(37); R.zeta_order()
36
sage: Zp(2).zeta_order()
2
"""
if (self.prime() == 2):
return 2
else:
return self.prime() - 1
def plot(self, max_points=2500, **args):
r"""
Create a visualization of this `p`-adic ring as a fractal
similar to a generalization of the Sierpi\'nski
triangle.
The resulting image attempts to capture the
algebraic and topological characteristics of `\mathbb{Z}_p`.
INPUT:
- ``max_points`` -- the maximum number or points to plot,
which controls the depth of recursion (default 2500)
- ``**args`` -- color, size, etc. that are passed to the
underlying point graphics objects
REFERENCES:
- Cuoco, A. ''Visualizing the `p`-adic Integers'', The
American Mathematical Monthly, Vol. 98, No. 4 (Apr., 1991),
pp. 355-364
EXAMPLES::
sage: Zp(3).plot()
Graphics object consisting of 1 graphics primitive
sage: Zp(5).plot(max_points=625)
Graphics object consisting of 1 graphics primitive
sage: Zp(23).plot(rgbcolor=(1,0,0))
Graphics object consisting of 1 graphics primitive
"""
if 'pointsize' not in args:
args['pointsize'] = 1
from sage.misc.mrange import cartesian_product_iterator
from sage.rings.real_double import RDF
from sage.plot.all import points, circle, Graphics
p = self.prime()
phi = 2*RDF.pi()/p
V = RDF**2
vs = [V([(phi*t).sin(), (phi*t).cos()]) for t in range(p)]
all = []
depth = max(RDF(max_points).log(p).floor(), 1)
scale = min(RDF(1.5/p), 1/RDF(3))
pts = [vs]*depth
if depth == 1 and 23 < p < max_points:
extras = int(max_points/p)
if p/extras > 5:
pts = [vs]*depth + [vs[::extras]]
for digits in cartesian_product_iterator(pts):
p = sum([v * scale**n for n, v in enumerate(digits)])
all.append(tuple(p))
g = points(all, **args)
# Set default plotting options
g.axes(False)
g.set_aspect_ratio(1)
return g