\(p\)-adic Base Generic#

A superclass for implementations of \(\ZZ_p\) and \(\QQ_p\).

AUTHORS:

  • David Roe

class sage.rings.padics.padic_base_generic.pAdicBaseGeneric(p, prec, print_mode, names, element_class)[source]#

Bases: pAdicGeneric

Initialization

absolute_discriminant()[source]#

Returns the absolute discriminant of this \(p\)-adic ring

EXAMPLES:

sage: Zp(5).absolute_discriminant()
1

>>> from sage.all import *
>>> Zp(Integer(5)).absolute_discriminant()
1
discriminant(K=None)[source]#

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

>>> from sage.all import *
>>> Zp(Integer(5)).discriminant()
1
exact_field()[source]#

Returns the rational field.

For compatibility with extensions of p-adics.

EXAMPLES:

sage: Zp(5).exact_field()
Rational Field

>>> from sage.all import *
>>> Zp(Integer(5)).exact_field()
Rational Field
exact_ring()[source]#

Returns the integer ring.

EXAMPLES:

sage: Zp(5).exact_ring()
Integer Ring

>>> from sage.all import *
>>> Zp(Integer(5)).exact_ring()
Integer Ring
gen(n=0)[source]#

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)

>>> from sage.all import *
>>> R = Zp(Integer(5)); R.gen()
5 + O(5^21)
has_pth_root()[source]#

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

>>> from sage.all import *
>>> Zp(Integer(2)).has_pth_root()
True
>>> Zp(Integer(17)).has_pth_root()
False
has_root_of_unity(n)[source]#

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

>>> from sage.all import *
>>> R=Zp(Integer(37))
>>> R.has_root_of_unity(Integer(12))
True
>>> R.has_root_of_unity(Integer(11))
False
is_abelian()[source]#

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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(10),'fixed-mod'); R.is_abelian()
True
is_isomorphic(ring)[source]#

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

>>> from sage.all import *
>>> R = Zp(Integer(5), Integer(15), print_mode='digits'); S = Zp(Integer(5), Integer(44), print_max_terms=Integer(4)); R.is_isomorphic(S)
True
is_normal()[source]#

Returns whether or not this is a normal extension, i.e. True.

EXAMPLES:

sage: R = Zp(3, 10,'fixed-mod'); R.is_normal()
True

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(10),'fixed-mod'); R.is_normal()
True
modulus(exact=False)[source]#

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

>>> from sage.all import *
>>> Zp(Integer(5)).modulus(exact=True)
x
plot(max_points=2500, **args)[source]#

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()                                                          # needs sage.plot
Graphics object consisting of 1 graphics primitive
sage: Zp(5).plot(max_points=625)                                            # needs sage.plot
Graphics object consisting of 1 graphics primitive
sage: Zp(23).plot(rgbcolor=(1,0,0))                                         # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> Zp(Integer(3)).plot()                                                          # needs sage.plot
Graphics object consisting of 1 graphics primitive
>>> Zp(Integer(5)).plot(max_points=Integer(625))                                            # needs sage.plot
Graphics object consisting of 1 graphics primitive
>>> Zp(Integer(23)).plot(rgbcolor=(Integer(1),Integer(0),Integer(0)))                                         # needs sage.plot
Graphics object consisting of 1 graphics primitive
uniformizer()[source]#

Returns a uniformizer for this ring.

EXAMPLES:

sage: R = Zp(3,5,'fixed-mod', 'series')
sage: R.uniformizer()
3

>>> from sage.all import *
>>> R = Zp(Integer(3),Integer(5),'fixed-mod', 'series')
>>> R.uniformizer()
3
uniformizer_pow(n)[source]#

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

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.uniformizer_pow(Integer(5))
5^5 + O(5^25)
>>> R.uniformizer_pow(infinity)
0
zeta(n=None)[source]#

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)

>>> from sage.all import *
>>> R = Zp(Integer(37),Integer(5))
>>> R.zeta(Integer(12))
8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
zeta_order()[source]#

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

>>> from sage.all import *
>>> R = Zp(Integer(37)); R.zeta_order()
36
>>> Zp(Integer(2)).zeta_order()
2