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)[source]

Bases: CommutativeRing

Initialize self.

EXAMPLES:

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

>>> from sage.all import *
>>> R = Zp(Integer(5))  # indirect doctest
>>> R.precision_cap()
20

R = Zp(5)  # indirect doctest
R.precision_cap()

In 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

>>> from sage.all import *
>>> TestSuite(R).run()                                                    # needs sage.geometry.polyhedron
>>> K = Qp(Integer(7))
>>> TestSuite(K).run()                                                    # needs sage.geometry.polyhedron

TestSuite(R).run()                                                    # needs sage.geometry.polyhedron
K = Qp(7)
TestSuite(K).run()                                                    # needs sage.geometry.polyhedron
absolute_degree()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.absolute_degree()                                                   # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.absolute_degree()                                                   # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.absolute_degree()                                                   # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.absolute_degree()                                                   # needs sage.libs.ntl
absolute_e()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.absolute_e()                                                        # needs sage.libs.ntl
1

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.absolute_e()                                                        # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.absolute_e()                                                        # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.absolute_e()                                                        # needs sage.libs.ntl
absolute_f()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.absolute_f()                                                        # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.absolute_f()                                                        # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.absolute_f()                                                        # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.absolute_f()                                                        # needs sage.libs.ntl
absolute_inertia_degree()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.absolute_inertia_degree()                                           # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.absolute_inertia_degree()                                           # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.absolute_inertia_degree()                                           # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.absolute_inertia_degree()                                           # needs sage.libs.ntl
absolute_ramification_index()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.absolute_ramification_index()                                       # needs sage.libs.ntl
1

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.absolute_ramification_index()                                       # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.absolute_ramification_index()                                       # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.absolute_ramification_index()                                       # needs sage.libs.ntl
change(**kwds)[source]

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 – boolean

  • print_sep – string

  • print_alphabet – dictionary

  • show_prec – boolean

  • check – boolean

  • 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 – boolean; 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

>>> from sage.all import *
>>> Zp(Integer(5)).change(prec=Integer(40))
5-adic Ring with capped relative precision 40

Zp(5).change(prec=40)

or the precision type:

sage: Zp(5).change(type='capped-abs')
5-adic Ring with capped absolute precision 20

>>> from sage.all import *
>>> Zp(Integer(5)).change(type='capped-abs')
5-adic Ring with capped absolute precision 20

Zp(5).change(type='capped-abs')

or even the prime:

sage: ZpCA(3).change(p=17)
17-adic Ring with capped absolute precision 20

>>> from sage.all import *
>>> ZpCA(Integer(3)).change(p=Integer(17))
17-adic Ring with capped absolute precision 20

ZpCA(3).change(p=17)

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

>>> from sage.all import *
>>> ZpCA(Integer(3)).change(field=True)
3-adic Field with capped relative precision 20

ZpCA(3).change(field=True)

You can also change print modes:

sage: R = Zp(5).change(prec=5, print_mode='digits')
sage: repr(~R(17))
'...13403'

>>> from sage.all import *
>>> R = Zp(Integer(5)).change(prec=Integer(5), print_mode='digits')
>>> repr(~R(Integer(17)))
'...13403'

R = Zp(5).change(prec=5, print_mode='digits')
repr(~R(17))

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'

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> S = ZZ['x']; (x,) = S._first_ngens(1)
>>> W = Zp(Integer(3)).extension(x**Integer(4) + Integer(9)*x**Integer(2) + Integer(3)*x - Integer(3), names=('w',)); (w,) = W._first_ngens(1)
>>> W.print_mode()
'series'
>>> W.change(print_mode='digits').print_mode()
'digits'

# needs sage.libs.ntl
S.<x> = ZZ[]
W.<w> = Zp(3).extension(x^4 + 9*x^2 + 3*x - 3)
W.print_mode()
W.change(print_mode='digits').print_mode()

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

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> K = QqFP(Integer(125), prec=Integer(4), names=('a',)); (a,) = K._first_ngens(1)
>>> K.change(q=Integer(64))
2-adic Unramified Extension Field in a defined by x^6 + x^4 + x^3 + x + 1
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K.change(modulus = x**Integer(2) - x + Integer(2), print_pos=False)
5-adic Unramified Extension Field in a defined by x^2 - x + 2

# needs sage.libs.flint
K.<a> = QqFP(125, prec=4)
K.change(q=64)
R.<x> = QQ[]
K.change(modulus = x^2 - x + 2, print_pos=False)

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

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

K.change(names='b')                                                   # needs sage.libs.flint

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

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> Kup = K.change(prec=Integer(8)); Kup
5-adic Unramified Extension Field in a defined by x^3 + 3*x + 3
>>> Kup.precision_cap()
8
>>> Kup.base_ring()
5-adic Field with floating precision 8

# needs sage.libs.flint
Kup = K.change(prec=8); Kup
Kup.precision_cap()
Kup.base_ring()

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

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

# needs sage.libs.flint
Kdown = K.change(prec=2); Kdown
Kdown.precision_cap()
Kdown.base_ring()

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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> x = polygen(ZZ)
>>> R = Zp(Integer(5)).extension(x**Integer(2) + Integer(2), names=('a',)); (a,) = R._first_ngens(1)
>>> S = R.change(p=Integer(7))
>>> S.defining_polynomial(exact=True)
x^2 + 2
>>> A = Zp(Integer(5))['y']; (y,) = A._first_ngens(1)
>>> R = Zp(Integer(5)).extension(y**Integer(2) + Integer(2), names=('a',)); (a,) = R._first_ngens(1)
>>> S = R.change(p=Integer(7))
>>> S.defining_polynomial(exact=True)
y^2 + 2

# needs sage.libs.ntl
x = polygen(ZZ)
R.<a> = Zp(5).extension(x^2 + 2)
S = R.change(p=7)
S.defining_polynomial(exact=True)
A.<y> = Zp(5)[]
R.<a> = Zp(5).extension(y^2 + 2)
S = R.change(p=7)
S.defining_polynomial(exact=True)
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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> R = Zq(Integer(5)**Integer(3), names=('a',)); (a,) = R._first_ngens(1)
>>> S = R.change(prec=Integer(50))
>>> S.defining_polynomial(exact=True)
x^3 + 3*x + 3

# needs sage.libs.ntl
R.<a> = Zq(5^3)
S = R.change(prec=50)
S.defining_polynomial(exact=True)
>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> R = Zq(Integer(5)**Integer(3), names=('a',)); (a,) = R._first_ngens(1)
>>> S = R.change(prec=Integer(50))
>>> S.defining_polynomial(exact=True)
x^3 + 3*x + 3

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

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)

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

R = ZpLC(37, (8,11))
S = R.change(label = "change"); S
S.change(label = "new")
defining_polynomial(var='x', exact=False)[source]

Return the defining polynomial of this local ring.

INPUT:

  • var – string (default: 'x'); the name of the variable

  • exact – 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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(3), 'fixed-mod')

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

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

R = Zp(3, 3, 'fixed-mod')
R.defining_polynomial().parent()
R.defining_polynomial('foo')
R.defining_polynomial(exact=True).parent()
degree()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.degree()                                                            # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.degree()                                                            # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.degree()                                                            # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.degree()                                                            # needs sage.libs.ntl
e()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.e()                                                                 # needs sage.libs.ntl
1

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.e()                                                                 # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.e()                                                                 # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.e()                                                                 # needs sage.libs.ntl
ext(*args, **kwds)[source]

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)

>>> from sage.all import *
>>> A = Zp(Integer(7),Integer(10))
>>> S = A['x']; (x,) = S._first_ngens(1)# needs sage.libs.ntl
>>> B = A.ext(x**Integer(2) + Integer(7), names=('t',)); (t,) = B._first_ngens(1)# needs sage.libs.ntl
>>> B.uniformiser()                                                       # needs sage.libs.ntl
t + O(t^21)

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

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.f()                                                                 # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.f()                                                                 # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.f()                                                                 # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.f()                                                                 # needs sage.libs.ntl
ground_ring()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(5), 'fixed-mod')
>>> S = Zp(Integer(3), Integer(4), 'fixed-mod')
>>> R.ground_ring() is R
True
>>> S.ground_ring() is R
False

R = Zp(3, 5, 'fixed-mod')
S = Zp(3, 4, 'fixed-mod')
R.ground_ring() is R
S.ground_ring() is R
ground_ring_of_tower()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.ground_ring_of_tower()
5-adic Ring with capped relative precision 20

R = Zp(5)
R.ground_ring_of_tower()
inertia_degree()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.inertia_degree()                                                    # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.inertia_degree()                                                    # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.inertia_degree()                                                    # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.inertia_degree()                                                    # needs sage.libs.ntl
inertia_subring()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.inertia_subring()
5-adic Ring with capped relative precision 20

R = Zp(5)
R.inertia_subring()
is_capped_absolute()[source]

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)

>>> from sage.all import *
>>> R = ZpCA(Integer(5), Integer(15))
>>> R.is_capped_absolute()
True
>>> R(Integer(5)**Integer(7))
5^7 + O(5^15)
>>> S = Zp(Integer(5), Integer(15))
>>> S.is_capped_absolute()
False
>>> S(Integer(5)**Integer(7))
5^7 + O(5^22)

R = ZpCA(5, 15)
R.is_capped_absolute()
R(5^7)
S = Zp(5, 15)
S.is_capped_absolute()
S(5^7)
is_capped_relative()[source]

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)

>>> from sage.all import *
>>> R = ZpCA(Integer(5), Integer(15))
>>> R.is_capped_relative()
False
>>> R(Integer(5)**Integer(7))
5^7 + O(5^15)
>>> S = Zp(Integer(5), Integer(15))
>>> S.is_capped_relative()
True
>>> S(Integer(5)**Integer(7))
5^7 + O(5^22)

R = ZpCA(5, 15)
R.is_capped_relative()
R(5^7)
S = Zp(5, 15)
S.is_capped_relative()
S(5^7)
is_exact()[source]

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

EXAMPLES:

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

>>> from sage.all import *
>>> R = Zp(Integer(5), Integer(3), 'fixed-mod'); R.is_exact()
False

R = Zp(5, 3, 'fixed-mod'); R.is_exact()
is_fixed_mod()[source]

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)

>>> from sage.all import *
>>> R = ZpFM(Integer(5),Integer(15))
>>> R.is_fixed_mod()
True
>>> R(Integer(5)**Integer(7),absprec=Integer(9))
5^7
>>> S = ZpCA(Integer(5), Integer(15))
>>> S.is_fixed_mod()
False
>>> S(Integer(5)**Integer(7),absprec=Integer(9))
5^7 + O(5^9)

R = ZpFM(5,15)
R.is_fixed_mod()
R(5^7,absprec=9)
S = ZpCA(5, 15)
S.is_fixed_mod()
S(5^7,absprec=9)
is_floating_point()[source]

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

>>> from sage.all import *
>>> R = ZpCR(Integer(5), Integer(15))
>>> R.is_floating_point()
False
>>> R(Integer(5)**Integer(7))
5^7 + O(5^22)
>>> S = ZpFP(Integer(5), Integer(15))
>>> S.is_floating_point()
True
>>> S(Integer(5)**Integer(7))
5^7

R = ZpCR(5, 15)
R.is_floating_point()
R(5^7)
S = ZpFP(5, 15)
S.is_floating_point()
S(5^7)
is_lattice_prec()[source]

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)

>>> from sage.all import *
>>> R = ZpCR(Integer(5), Integer(15))
>>> R.is_lattice_prec()
False
>>> x = R(Integer(25), Integer(8))
>>> x - x
O(5^8)
>>> S = ZpLC(Integer(5), Integer(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.
>>> S.is_lattice_prec()
True
>>> x = S(Integer(25), Integer(8))
>>> x - x
O(5^30)

R = ZpCR(5, 15)
R.is_lattice_prec()
x = R(25, 8)
x - x
S = ZpLC(5, 15)
S.is_lattice_prec()
x = S(25, 8)
x - x
is_relaxed()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.is_relaxed()
False

R = Zp(5)
R.is_relaxed()
maximal_unramified_subextension()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.maximal_unramified_subextension()
5-adic Ring with capped relative precision 20

R = Zp(5)
R.maximal_unramified_subextension()
precision_cap()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(10),'fixed-mod'); R.precision_cap()
10
>>> R = Zp(Integer(3), Integer(10),'capped-rel'); R.precision_cap()
10
>>> R = Zp(Integer(3), Integer(10),'capped-abs'); R.precision_cap()
10

R = Zp(3, 10,'fixed-mod'); R.precision_cap()
R = Zp(3, 10,'capped-rel'); R.precision_cap()
R = Zp(3, 10,'capped-abs'); R.precision_cap()

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()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.ramification_index()                                                # needs sage.libs.ntl
1

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.ramification_index()                                                # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.ramification_index()                                                # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.ramification_index()                                                # needs sage.libs.ntl
relative_degree()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.relative_degree()                                                   # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.relative_degree()                                                   # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.relative_degree()                                                   # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.relative_degree()                                                   # needs sage.libs.ntl
relative_e()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.relative_e()                                                        # needs sage.libs.ntl
1

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.relative_e()                                                        # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.relative_e()                                                        # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.relative_e()                                                        # needs sage.libs.ntl
relative_f()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.relative_f()                                                        # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.relative_f()                                                        # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.relative_f()                                                        # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.relative_f()                                                        # needs sage.libs.ntl
relative_inertia_degree()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.relative_inertia_degree()                                           # needs sage.libs.ntl
5

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.relative_inertia_degree()                                           # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.relative_inertia_degree()                                           # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.relative_inertia_degree()                                           # needs sage.libs.ntl
relative_ramification_index()[source]

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> K.relative_ramification_index()                                       # needs sage.libs.ntl
1

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(3)).extension(x**Integer(2) - Integer(3), names=('pi',)); (pi,) = L._first_ngens(1)# needs sage.libs.ntl
>>> L.relative_ramification_index()                                       # needs sage.libs.ntl
2

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.relative_ramification_index()                                       # needs sage.libs.ntl
R.<x> = QQ[]
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.relative_ramification_index()                                       # needs sage.libs.ntl
residue_characteristic()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(5), 'capped-rel'); R.residue_characteristic()
3

R = Zp(3, 5, 'capped-rel'); R.residue_characteristic()
uniformiser()[source]

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)

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.uniformiser()
5 + O(5^21)
>>> A = Zp(Integer(7),Integer(10))
>>> S = A['x']; (x,) = S._first_ngens(1)# needs sage.libs.ntl
>>> B = A.ext(x**Integer(2)+Integer(7), names=('t',)); (t,) = B._first_ngens(1)# needs sage.libs.ntl
>>> B.uniformiser()                                                       # needs sage.libs.ntl
t + O(t^21)

R = Zp(5)
R.uniformiser()
A = Zp(7,10)
S.<x> = A[]                                                           # needs sage.libs.ntl
B.<t> = A.ext(x^2+7)                                                  # needs sage.libs.ntl
B.uniformiser()                                                       # needs sage.libs.ntl
uniformiser_pow(n)[source]

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)

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R.uniformiser_pow(Integer(5))
5^5 + O(5^25)

R = Zp(5)
R.uniformiser_pow(5)