\(p\)-Adic Generic Nodes

This file contains a bunch of intermediate classes for the \(p\)-adic parents, allowing a function to be implemented at the right level of generality.

AUTHORS:

• David Roe

class sage.rings.padics.generic_nodes.CappedAbsoluteGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

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()

TESTS:

sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False

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)

class sage.rings.padics.generic_nodes.CappedRelativeFieldGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.CappedRelativeGeneric

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()

TESTS:

sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False

class sage.rings.padics.generic_nodes.CappedRelativeGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

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()

TESTS:

sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False

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)

class sage.rings.padics.generic_nodes.CappedRelativeRingGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.CappedRelativeGeneric

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()

TESTS:

sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False

class sage.rings.padics.generic_nodes.FixedModGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

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()

TESTS:

sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False

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)

sage.rings.padics.generic_nodes.is_pAdicField(R)

Returns True if and only if R is a \(p\)-adic field.

EXAMPLES:

sage: is_pAdicField(Zp(17))
False
sage: is_pAdicField(Qp(17))
True

sage.rings.padics.generic_nodes.is_pAdicRing(R)

Returns True if and only if R is a \(p\)-adic ring (not a field).

EXAMPLES:

sage: is_pAdicRing(Zp(5))
True
sage: is_pAdicRing(RR)
False

class sage.rings.padics.generic_nodes.pAdicCappedAbsoluteRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.CappedAbsoluteGeneric

Initialization.

INPUTS:

- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring

EXAMPLES:

sage: R = Zp(17) #indirect doctest

class sage.rings.padics.generic_nodes.pAdicCappedRelativeFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicFieldGeneric, sage.rings.padics.generic_nodes.CappedRelativeFieldGeneric

Initialization.

INPUTS:

- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring

EXAMPLES:

sage: R = Zp(17) #indirect doctest

class sage.rings.padics.generic_nodes.pAdicCappedRelativeRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.CappedRelativeRingGeneric

Initialization.

INPUTS:

- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring

EXAMPLES:

sage: R = Zp(17) #indirect doctest

class sage.rings.padics.generic_nodes.pAdicFieldBaseGeneric(p, prec, print_mode, names, element_class)

Bases: sage.rings.padics.padic_base_generic.pAdicBaseGeneric, sage.rings.padics.generic_nodes.pAdicFieldGeneric

Initialization

TESTS:

sage: R = Zp(5) #indirect doctest

composite(subfield1, subfield2)

Returns the composite of two subfields of self, i.e., the largest subfield containing both

INPUT:

• self – a \(p\)-adic field
• subfield1 – a subfield
• subfield2 – a subfield

OUTPUT:

• the composite of subfield1 and subfield2

EXAMPLES:

sage: K = Qp(17); K.composite(K, K) is K
True

construction()

Returns the functorial construction of self, namely, completion of the rational numbers with respect a given prime.

Also preserves other information that makes this field unique (e.g. precision, rounding, print mode).

EXAMPLE:

sage: K = Qp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
Rational Field
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True

subfield(list)

Returns the subfield generated by the elements in list

INPUT:

• self – a \(p\)-adic field
• list – a list of elements of self

OUTPUT:

• the subfield of self generated by the elements of list

EXAMPLES:

sage: K = Qp(17); K.subfield([K(17), K(1827)]) is K
True

subfields_of_degree(n)

Returns the number of subfields of self of degree \(n\)

INPUT:

• self – a \(p\)-adic field
• n – an integer

OUTPUT:

• integer – the number of subfields of degree n over self.base_ring()

EXAMPLES:

sage: K = Qp(17)
sage: K.subfields_of_degree(1)
1

class sage.rings.padics.generic_nodes.pAdicFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.padic_generic.pAdicGeneric, sage.rings.ring.Field

Initialization.

INPUTS:

- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring

EXAMPLES:

sage: R = Zp(17) #indirect doctest

class sage.rings.padics.generic_nodes.pAdicFixedModRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.FixedModGeneric

Initialization.

INPUTS:

- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring

EXAMPLES:

sage: R = Zp(17) #indirect doctest

class sage.rings.padics.generic_nodes.pAdicRingBaseGeneric(p, prec, print_mode, names, element_class)

Bases: sage.rings.padics.padic_base_generic.pAdicBaseGeneric, sage.rings.padics.generic_nodes.pAdicRingGeneric

Initialization

TESTS:

sage: R = Zp(5) #indirect doctest

construction()

Returns the functorial construction of self, namely, completion of the rational numbers with respect a given prime.

Also preserves other information that makes this field unique (e.g. precision, rounding, print mode).

EXAMPLE:

sage: K = Zp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
Integer Ring
sage: c(L)
17-adic Ring with capped relative precision 8
sage: K == c(L)
True

random_element(algorithm='default')

Returns a random element of self, optionally using the algorithm argument to decide how it generates the element. Algorithms currently implemented:

• default: Choose \(a_i\), \(i >= 0\), randomly between \(0\) and \(p-1\) until a nonzero choice is made. Then continue choosing \(a_i\) randomly between \(0\) and \(p-1\) until we reach precision_cap, and return \(\sum a_i p^i\).

EXAMPLES:

sage: Zp(5,6).random_element()
3 + 3*5 + 2*5^2 + 3*5^3 + 2*5^4 + 5^5 + O(5^6)
sage: ZpCA(5,6).random_element()
4*5^2 + 5^3 + O(5^6)
sage: ZpFM(5,6).random_element()
2 + 4*5^2 + 2*5^4 + 5^5 + O(5^6)

class sage.rings.padics.generic_nodes.pAdicRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.padic_generic.pAdicGeneric, sage.rings.ring.EuclideanDomain

Initialization.

INPUTS:

- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring

EXAMPLES:

sage: R = Zp(17) #indirect doctest

is_field(proof=True)

Returns whether this ring is actually a field, ie False.

EXAMPLES:

sage: Zp(5).is_field()
False

krull_dimension()

Returns the Krull dimension of self, i.e. 1

INPUT:

• self – a \(p\)-adic ring

OUTPUT:

• the Krull dimension of self. Since self is a \(p\)-adic ring, this is 1.

EXAMPLES:

sage: Zp(5).krull_dimension()
1