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: CommutativeRing

Initialize self.

EXAMPLES:

sage: R = Zp(5)  # indirect doctest
sage: R.precision_cap()
20

In github issue #14084, the category framework has been implemented for p-adic rings:

sage: TestSuite(R).run()                                                    # needs sage.geometry.polyhedron
sage: K = Qp(7)
sage: TestSuite(K).run()                                                    # needs sage.geometry.polyhedron

absolute_degree()#

Return the degree of this extension over the prime p-adic field/ring.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.absolute_degree()                                                   # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.absolute_degree()                                                   # needs sage.libs.ntl
2

absolute_e()#

Return the absolute ramification index of this ring/field.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.absolute_e()                                                        # needs sage.libs.ntl
1

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.absolute_e()                                                        # needs sage.libs.ntl
2

absolute_f()#

Return the degree of the residue field of this ring/field over its prime subfield.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.absolute_f()                                                        # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.absolute_f()                                                        # needs sage.libs.ntl
1

absolute_inertia_degree()#

Return the degree of the residue field of this ring/field over its prime subfield.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.absolute_inertia_degree()                                           # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.absolute_inertia_degree()                                           # needs sage.libs.ntl
1

absolute_ramification_index()#

Return the absolute ramification index of this ring/field.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.absolute_ramification_index()                                       # needs sage.libs.ntl
1

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.absolute_ramification_index()                                       # needs sage.libs.ntl
2

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 type
p – the prime of the ground ring. Defining polynomials
will be converted to the new base rings.
print_mode – string
print_pos – bool
print_sep – string
print_alphabet – dict
show_prec – bool
check – bool
label – string (only for lattice precision)

The following arguments are only applied to the top ring in the tower:

var_name – string
res_name – string
unram_name – string
ram_name – string
names – string
modulus – 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. If True, 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: # needs sage.libs.ntl
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: # needs sage.libs.flint
sage: K.<a> = QqFP(125, prec=4)
sage: K.change(q=64)
2-adic Unramified Extension Field in a defined by x^6 + x^4 + x^3 + x + 1
sage: R.<x> = QQ[]
sage: K.change(modulus = x^2 - x + 2, print_pos=False)
5-adic Unramified Extension Field in a defined by x^2 - x + 2

and variable names:

sage: K.change(names='b')                                                   # needs sage.libs.flint
5-adic Unramified Extension Field in b defined by x^3 + 3*x + 3

and precision:

sage: # needs sage.libs.flint
sage: Kup = K.change(prec=8); Kup
5-adic Unramified Extension Field in a defined by x^3 + 3*x + 3
sage: Kup.precision_cap()
8
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: # needs sage.libs.flint
sage: Kdown = K.change(prec=2); Kdown
5-adic Unramified Extension Field in a defined by x^3 + 3*x + 3
sage: Kdown.precision_cap()
2
sage: Kdown.base_ring()
5-adic Field with floating precision 4

Changing the prime works for extensions:

sage: # needs sage.libs.ntl
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: # needs sage.libs.ntl
sage: R.<a> = Zq(5^3)
sage: S = R.change(prec=50)
sage: S.defining_polynomial(exact=True)
x^3 + 3*x + 3

Changing label for lattice precision (the precision lattice is not copied):

sage: R = ZpLC(37, (8,11))
sage: S = R.change(label = "change"); S
37-adic Ring with lattice-cap precision (label: change)
sage: S.change(label = "new")
37-adic Ring with lattice-cap precision (label: new)

defining_polynomial(var='x', exact=False)#

Return the defining polynomial of this local ring

INPUT:

var – string (default: 'x'), the name of the variable
exact – a boolean (default: False), whether to return the underlying exact defining polynomial rather than the one with coefficients in the base ring.

OUTPUT:

The defining polynomial of this ring as an extension over its ground ring

EXAMPLES:

sage: R = Zp(3, 3, 'fixed-mod')

sage: R.defining_polynomial().parent()
Univariate Polynomial Ring in x over 3-adic Ring of fixed modulus 3^3
sage: R.defining_polynomial('foo')
foo

sage: R.defining_polynomial(exact=True).parent()
Univariate Polynomial Ring in x over Integer Ring

degree()#

Return the degree of this extension.

Raise an error if the base ring/field is itself an extension.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.degree()                                                            # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.degree()                                                            # needs sage.libs.ntl
2

e()#

Return the ramification index of this extension.

Raise an error if the base ring/field is itself an extension.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.e()                                                                 # needs sage.libs.ntl
1

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.e()                                                                 # needs sage.libs.ntl
2

ext(*args, **kwds)#

Construct an extension of self. See extension() for more details.

EXAMPLES:

sage: A = Zp(7,10)
sage: S.<x> = A[]                                                           # needs sage.libs.ntl
sage: B.<t> = A.ext(x^2 + 7)                                                # needs sage.libs.ntl
sage: B.uniformiser()                                                       # needs sage.libs.ntl
t + O(t^21)

f()#

Return the degree of the residual extension.

Raise an error if the base ring/field is itself an extension.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.f()                                                                 # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.f()                                                                 # needs sage.libs.ntl
1

ground_ring()#

Return 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

ground_ring_of_tower()#

Return 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

inertia_degree()#

Return the degree of the residual extension.

Raise an error if the base ring/field is itself an extension.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.inertia_degree()                                                    # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.inertia_degree()                                                    # needs sage.libs.ntl
1

inertia_subring()#

Return 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

is_capped_absolute()#

Return 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)

is_capped_relative()#

Return 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)

is_exact()#

Return whether this \(p\)-adic ring is exact, i.e. False.

EXAMPLES:

sage: R = Zp(5, 3, 'fixed-mod'); R.is_exact()
False

is_fixed_mod()#

Return 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
sage: S = ZpCA(5, 15)
sage: S.is_fixed_mod()
False
sage: S(5^7,absprec=9)
5^7 + O(5^9)

is_floating_point()#

Return 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

is_lattice_prec()#

Return whether this \(p\)-adic ring bounds precision using a lattice model.

In lattice precision, relationships between elements are stored in a precision object of the parent, which allows for optimal precision tracking at the cost of increased memory usage and runtime.

EXAMPLES:

sage: R = ZpCR(5, 15)
sage: R.is_lattice_prec()
False
sage: x = R(25, 8)
sage: x - x
O(5^8)
sage: S = ZpLC(5, 15)
doctest:...: FutureWarning: This class/method/function is marked as experimental. It, its functionality or its interface might change without a formal deprecation.
See https://github.com/sagemath/sage/issues/23505 for details.
sage: S.is_lattice_prec()
True
sage: x = S(25, 8)
sage: x - x
O(5^30)

is_relaxed()#

Return whether this \(p\)-adic ring bounds precision in a relaxed fashion.

In a relaxed ring, elements have mechanisms for computing themselves to greater precision.

EXAMPLES:

sage: R = Zp(5)
sage: R.is_relaxed()
False

maximal_unramified_subextension()#

Return 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

precision_cap()#

Return the precision cap for this ring.

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 class pAdicRingCappedAbsolute), each element has a precision that is tracked and is bounded above by self.precision_cap(). Rings with relative caps (e.g. the class pAdicRingCappedRelative) are the same except that the precision is the precision of the unit part of each element.

ramification_index()#

Return the ramification index of this extension.

Raise an error if the base ring/field is itself an extension.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.ramification_index()                                                # needs sage.libs.ntl
1

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.ramification_index()                                                # needs sage.libs.ntl
2

relative_degree()#

Return the degree of this extension over its base field/ring.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.relative_degree()                                                   # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.relative_degree()                                                   # needs sage.libs.ntl
2

relative_e()#

Return the ramification index of this extension over its base ring/field.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.relative_e()                                                        # needs sage.libs.ntl
1

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.relative_e()                                                        # needs sage.libs.ntl
2

relative_f()#

Return the degree of the residual extension over its base ring/field.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.relative_f()                                                        # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.relative_f()                                                        # needs sage.libs.ntl
1

relative_inertia_degree()#

Return the degree of the residual extension over its base ring/field.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.relative_inertia_degree()                                           # needs sage.libs.ntl
5

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.relative_inertia_degree()                                           # needs sage.libs.ntl
1

relative_ramification_index()#

Return the ramification index of this extension over its base ring/field.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.relative_ramification_index()                                       # needs sage.libs.ntl
1

sage: R.<x> = QQ[]
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.relative_ramification_index()                                       # needs sage.libs.ntl
2

residue_characteristic()#

Return the characteristic of self’s residue field.

INPUT:

self – a p-adic ring.

OUTPUT:

The characteristic of the residue field.

EXAMPLES:

sage: R = Zp(3, 5, 'capped-rel'); R.residue_characteristic()
3

uniformiser()#

Return 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[]                                                           # needs sage.libs.ntl
sage: B.<t> = A.ext(x^2+7)                                                  # needs sage.libs.ntl
sage: B.uniformiser()                                                       # needs sage.libs.ntl
t + O(t^21)

uniformiser_pow(n)#

Return 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)