\(p\)-Adic Base Generic¶
A superclass for implementations of \(\ZZ_p\) and \(\QQ_p\).
AUTHORS:
David Roe
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class
sage.rings.padics.padic_base_generic.
pAdicBaseGeneric
(p, prec, print_mode, names, element_class)¶ Bases:
sage.rings.padics.padic_generic.pAdicGeneric
Initialization
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absolute_discriminant
()¶ Returns the absolute discriminant of this \(p\)-adic ring
EXAMPLES:
sage: Zp(5).absolute_discriminant() 1
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discriminant
(K=None)¶ Returns the discriminant of this \(p\)-adic ring over
K
INPUT:
self
– a \(p\)-adic ringK
– a sub-ring ofself
orNone
(default:None
)
OUTPUT:
integer – the discriminant of this ring over
K
(or the absolute discriminant ifK
isNone
)
EXAMPLES:
sage: Zp(5).discriminant() 1
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exact_field
()¶ Returns the rational field.
For compatibility with extensions of p-adics.
EXAMPLES:
sage: Zp(5).exact_field() Rational Field
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exact_ring
()¶ Returns the integer ring.
EXAMPLES:
sage: Zp(5).exact_ring() Integer Ring
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gen
(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)
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has_pth_root
()¶ 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
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has_root_of_unity
(n)¶ Returns whether or not \(\ZZ_p\) has a primitive \(n^{th}\) root of unity.
INPUT:
self
– a \(p\)-adic ringn
– an integer
OUTPUT:
boolean
– whetherself
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
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is_abelian
()¶ 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
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is_isomorphic
(ring)¶ Returns whether
self
andring
are isomorphic, i.e. whetherring
is an implementation of \(\ZZ_p\) for the same prime asself
.INPUT:
self
– a \(p\)-adic ringring
– a ring
OUTPUT:
boolean
– whetherring
is an implementation of ZZ_p` for the same prime asself
.
EXAMPLES:
sage: R = Zp(5, 15, print_mode='digits'); S = Zp(5, 44, print_max_terms=4); R.is_isomorphic(S) True
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is_normal
()¶ Returns whether or not this is a normal extension, i.e.
True
.EXAMPLES:
sage: R = Zp(3, 10,'fixed-mod'); R.is_normal() True
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modulus
(exact=False)¶ Returns the polynomial defining this extension.
For compatibility with extension fields; we define the modulus to be x-1.
INPUT:
exact
– boolean (defaultFalse
), whether to return a polynomial with integer entries.
EXAMPLES:
sage: Zp(5).modulus(exact=True) x
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plot
(max_points=2500, **args)¶ 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
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uniformizer
()¶ Returns a uniformizer for this ring.
EXAMPLES:
sage: R = Zp(3,5,'fixed-mod', 'series') sage: R.uniformizer() 3
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uniformizer_pow
(n)¶ Returns the
nth
power of the uniformizer ofself
(as an element ofself
).EXAMPLES:
sage: R = Zp(5) sage: R.uniformizer_pow(5) 5^5 + O(5^25) sage: R.uniformizer_pow(infinity) 0
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zeta
(n=None)¶ Returns a generator of the group of roots of unity.
INPUT:
self
– a \(p\)-adic ringn
– an integer orNone
(default:None
)
OUTPUT:
element
– a generator of the \(n^{th}\) roots of unity, or a generator of the full group of roots of unity ifn
isNone
EXAMPLES:
sage: R = Zp(37,5) sage: R.zeta(12) 8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
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zeta_order
()¶ 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
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