Local Generic¶
Superclass for \(p\)-adic and power series rings.
AUTHORS:
- David Roe
-
class
sage.rings.padics.local_generic.
LocalGeneric
(base, prec, names, element_class, category=None)¶ Bases:
sage.rings.ring.CommutativeRing
Initializes self.
EXAMPLES:
sage: R = Zp(5) #indirect doctest sage: R.precision_cap() 20
In trac ticket #14084, the category framework has been implemented for p-adic rings:
sage: TestSuite(R).run() sage: K = Qp(7) sage: TestSuite(K).run()
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change
(**kwds)¶ Return a new ring with changed attributes.
INPUT:
The following arguments are applied to every ring in the tower:
type
– string, the precision typep
– the prime of the ground ring. Defining polynomials- will be converted to the new base rings.
print_mode
– stringprint_pos
– boolprint_sep
– stringprint_alphabet
– dictshow_prec
– boolcheck
– bool
The following arguments are only applied to the top ring in the tower:
var_name
– stringres_name
– stringunram_name
– stringram_name
– stringnames
– stringmodulus
– polynomial
The following arguments have special behavior:
prec
– integer. If the precision is increased on an extension ring,- the precision on the base is increased as necessary (respecting ramification). If the precision is decreased, the precision of the base is unchanged.
field
– bool. IfTrue
, switch to a tower of fields via the fraction field.- If False, switch to a tower of rings of integers.
q
– prime power. Replace the initial unramified extension of \(\QQ_p\) or \(\ZZ_p\)- with an unramified extension of residue cardinality \(q\). If the initial extension is ramified, add in an unramified extension.
base
– ring or field. Use a specific base ring instead of recursively- calling
change()
down the tower.
See the
constructors
for more details on the meaning of these arguments.EXAMPLES:
We can use this method to change the precision:
sage: Zp(5).change(prec=40) 5-adic Ring with capped relative precision 40
or the precision type:
sage: Zp(5).change(type="capped-abs") 5-adic Ring with capped absolute precision 20
or even the prime:
sage: ZpCA(3).change(p=17) 17-adic Ring with capped absolute precision 20
You can switch between the ring of integers and its fraction field:
sage: ZpCA(3).change(field=True) 3-adic Field with capped relative precision 20
You can also change print modes:
sage: R = Zp(5).change(prec=5, print_mode='digits') sage: repr(~R(17)) '...13403'
Changing print mode to ‘digits’ works for Eisenstein extensions:
sage: S.<x> = ZZ[] sage: W.<w> = Zp(3).extension(x^4 + 9*x^2 + 3*x - 3) sage: W.print_mode() 'series' sage: W.change(print_mode='digits').print_mode() 'digits'
You can change extensions:
sage: K.<a> = QqFP(125, prec=4) sage: K.change(q=64) Unramified Extension in a defined by x^6 + x^4 + x^3 + x + 1 with floating precision 4 over 2-adic Field sage: R.<x> = QQ[] sage: K.change(modulus = x^2 - x + 2, print_pos=False) Unramified Extension in a defined by x^2 - x + 2 with floating precision 4 over 5-adic Field
and variable names:
sage: K.change(names='b') Unramified Extension in b defined by x^3 + 3*x + 3 with floating precision 4 over 5-adic Field
and precision:
sage: Kup = K.change(prec=8); Kup Unramified Extension in a defined by x^3 + 3*x + 3 with floating precision 8 over 5-adic Field sage: Kup.base_ring() 5-adic Field with floating precision 8
If you decrease the precision, the precision of the base stays the same:
sage: Kdown = K.change(prec=2); Kdown Unramified Extension in a defined by x^3 + 3*x + 3 with floating precision 2 over 5-adic Field sage: Kdown.precision_cap() 2 sage: Kdown.base_ring() 5-adic Field with floating precision 4
Changing the prime works for extensions:
sage: x = polygen(ZZ) sage: R.<a> = Zp(5).extension(x^2 + 2) sage: S = R.change(p=7) sage: S.defining_polynomial(exact=True) x^2 + 2 sage: A.<y> = Zp(5)[] sage: R.<a> = Zp(5).extension(y^2 + 2) sage: S = R.change(p=7) sage: S.defining_polynomial(exact=True) y^2 + 2
sage: R.<a> = Zq(5^3) sage: S = R.change(prec=50) sage: S.defining_polynomial(exact=True) x^3 + 3*x + 3
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defining_polynomial
(var='x')¶ Returns the defining polynomial of this local ring, i.e. just
x
.INPUT:
self
– a local ringvar
– string (default:'x'
) the name of the variable
OUTPUT:
- polynomial – the defining polynomial of this ring as an extension over its ground ring
EXAMPLES:
sage: R = Zp(3, 3, 'fixed-mod'); R.defining_polynomial('foo') (1 + O(3^3))*foo + (O(3^3))
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degree
()¶ Returns the degree of
self
over the ground ring, i.e. 1.INPUT:
self
– a local ring
OUTPUT:
- integer – the degree of this ring, i.e., 1
EXAMPLES:
sage: R = Zp(3, 10, 'capped-rel'); R.degree() 1
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e
(K=None)¶ Returns the ramification index over the ground ring: 1 unless overridden.
INPUT:
self
– a local ringK
– a subring ofself
(defaultNone
)
OUTPUT:
- integer – the ramification index of this ring: 1 unless overridden.
EXAMPLES:
sage: R = Zp(3, 5, 'capped-rel'); R.e() 1
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ext
(*args, **kwds)¶ Constructs an extension of self. See
extension
for more details.EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] sage: B.<t> = A.ext(x^2+7) sage: B.uniformiser() t + O(t^21)
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f
(K=None)¶ Returns the inertia degree over the ground ring: 1 unless overridden.
INPUT:
self
– a local ringK
– a subring (defaultNone
)
OUTPUT:
- integer – the inertia degree of this ring: 1 unless overridden.
EXAMPLES:
sage: R = Zp(3, 5, 'capped-rel'); R.f() 1
-
ground_ring
()¶ Returns
self
.Will be overridden by extensions.
INPUT:
self
– a local ring
OUTPUT:
- the ground ring of
self
, i.e., itself
EXAMPLES:
sage: R = Zp(3, 5, 'fixed-mod') sage: S = Zp(3, 4, 'fixed-mod') sage: R.ground_ring() is R True sage: S.ground_ring() is R False
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ground_ring_of_tower
()¶ Returns
self
.Will be overridden by extensions.
INPUT:
self
– a \(p\)-adic ring
OUTPUT:
- the ground ring of the tower for
self
, i.e., itself
EXAMPLES:
sage: R = Zp(5) sage: R.ground_ring_of_tower() 5-adic Ring with capped relative precision 20
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inertia_degree
(K=None)¶ Returns the inertia degree over
K
(defaults to the ground ring): 1 unless overridden.INPUT:
self
– a local ringK
– a subring ofself
(default None)
OUTPUT:
- integer – the inertia degree of this ring: 1 unless overridden.
EXAMPLES:
sage: R = Zp(3, 5, 'capped-rel'); R.inertia_degree() 1
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inertia_subring
()¶ Returns the inertia subring, i.e.
self
.INPUT:
self
– a local ring
OUTPUT:
- the inertia subring of self, i.e., itself
EXAMPLES:
sage: R = Zp(5) sage: R.inertia_subring() 5-adic Ring with capped relative precision 20
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is_capped_absolute
()¶ Returns whether this \(p\)-adic ring bounds precision in a capped absolute fashion.
The absolute precision of an element is the power of \(p\) modulo which that element is defined. In a capped absolute ring, the absolute precision of elements are bounded by a constant depending on the ring.
EXAMPLES:
sage: R = ZpCA(5, 15) sage: R.is_capped_absolute() True sage: R(5^7) 5^7 + O(5^15) sage: S = Zp(5, 15) sage: S.is_capped_absolute() False sage: S(5^7) 5^7 + O(5^22)
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is_capped_relative
()¶ Returns whether this \(p\)-adic ring bounds precision in a capped relative fashion.
The relative precision of an element is the power of \(p\) modulo which the unit part of that element is defined. In a capped relative ring, the relative precision of elements are bounded by a constant depending on the ring.
EXAMPLES:
sage: R = ZpCA(5, 15) sage: R.is_capped_relative() False sage: R(5^7) 5^7 + O(5^15) sage: S = Zp(5, 15) sage: S.is_capped_relative() True sage: S(5^7) 5^7 + O(5^22)
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is_exact
()¶ Returns whether this p-adic ring is exact, i.e. False.
- INPUT:
- self – a p-adic ring
- OUTPUT:
- boolean – whether self is exact, i.e. False.
EXAMPLES:
#sage: R = Zp(5, 3, 'lazy'); R.is_exact() #False sage: R = Zp(5, 3, 'fixed-mod'); R.is_exact() False
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is_finite
()¶ Returns whether this ring is finite, i.e.
False
.INPUT:
self
– a \(p\)-adic ring
OUTPUT:
- boolean – whether self is finite, i.e.,
False
EXAMPLES:
sage: R = Zp(3, 10,'fixed-mod'); R.is_finite() False
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is_fixed_mod
()¶ Returns whether this \(p\)-adic ring bounds precision in a fixed modulus fashion.
The absolute precision of an element is the power of \(p\) modulo which that element is defined. In a fixed modulus ring, the absolute precision of every element is defined to be the precision cap of the parent. This means that some operations, such as division by \(p\), don’t return a well defined answer.
EXAMPLES:
sage: R = ZpFM(5,15) sage: R.is_fixed_mod() True sage: R(5^7,absprec=9) 5^7 + O(5^15) sage: S = ZpCA(5, 15) sage: S.is_fixed_mod() False sage: S(5^7,absprec=9) 5^7 + O(5^9)
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is_floating_point
()¶ Returns whether this \(p\)-adic ring bounds precision in a floating point fashion.
The relative precision of an element is the power of \(p\) modulo which the unit part of that element is defined. In a floating point ring, elements do not store precision, but arithmetic operations truncate to a relative precision depending on the ring.
EXAMPLES:
sage: R = ZpCR(5, 15) sage: R.is_floating_point() False sage: R(5^7) 5^7 + O(5^22) sage: S = ZpFP(5, 15) sage: S.is_floating_point() True sage: S(5^7) 5^7
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is_lazy
()¶ Returns whether this \(p\)-adic ring bounds precision in a lazy fashion.
In a lazy ring, elements have mechanisms for computing themselves to greater precision.
EXAMPLES:
sage: R = Zp(5) sage: R.is_lazy() False
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maximal_unramified_subextension
()¶ Returns the maximal unramified subextension.
INPUT:
self
– a local ring
OUTPUT:
- the maximal unramified subextension of
self
EXAMPLES:
sage: R = Zp(5) sage: R.maximal_unramified_subextension() 5-adic Ring with capped relative precision 20
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precision_cap
()¶ Returns the precision cap for
self
.INPUT:
self
– a local ring
OUTPUT:
- integer –
self
’s precision cap
EXAMPLES:
sage: R = Zp(3, 10,'fixed-mod'); R.precision_cap() 10 sage: R = Zp(3, 10,'capped-rel'); R.precision_cap() 10 sage: R = Zp(3, 10,'capped-abs'); R.precision_cap() 10
Note
This will have different meanings depending on the type of local ring. For fixed modulus rings, all elements are considered modulo
self.prime()^self.precision_cap()
. For rings with an absolute cap (i.e. the classpAdicRingCappedAbsolute
), each element has a precision that is tracked and is bounded above byself.precision_cap()
. Rings with relative caps (e.g. the classpAdicRingCappedRelative
) are the same except that the precision is the precision of the unit part of each element. For lazy rings, this gives the initial precision to which elements are computed.
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ramification_index
(K=None)¶ Returns the ramification index over the ground ring: 1 unless overridden.
INPUT:
self
– a local ring
OUTPUT:
- integer – the ramification index of this ring: 1 unless overridden.
EXAMPLES:
sage: R = Zp(3, 5, 'capped-rel'); R.ramification_index() 1
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residue_characteristic
()¶ Returns the characteristic of
self
’s residue field.INPUT:
self
– a p-adic ring.
OUTPUT:
- integer – the characteristic of the residue field.
EXAMPLES:
sage: R = Zp(3, 5, 'capped-rel'); R.residue_characteristic() 3
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residue_class_degree
(K=None)¶ Returns the inertia degree over the ground ring: 1 unless overridden.
INPUT:
self
– a local ringK
– a subring (defaultNone
)
OUTPUT:
- integer – the inertia degree of this ring: 1 unless overridden.
EXAMPLES:
sage: R = Zp(3, 5, 'capped-rel'); R.residue_class_degree() 1
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uniformiser
()¶ Returns a uniformiser for
self
, ie a generator for the unique maximal ideal.EXAMPLES:
sage: R = Zp(5) sage: R.uniformiser() 5 + O(5^21) sage: A = Zp(7,10) sage: S.<x> = A[] sage: B.<t> = A.ext(x^2+7) sage: B.uniformiser() t + O(t^21)
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uniformiser_pow
(n)¶ Returns the \(n`th power of the uniformiser of ``self`\) (as an element of
self
).EXAMPLES:
sage: R = Zp(5) sage: R.uniformiser_pow(5) 5^5 + O(5^25)
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