$p$ -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)¶

Bases: sage.rings.padics.local_generic.LocalGeneric

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)¶: Bases: sage.rings.padics.generic_nodes.CappedRelativeGeneric

class sage.rings.padics.generic_nodes.CappedRelativeGeneric(base, prec, names, element_class)¶

Bases: sage.rings.padics.local_generic.LocalGeneric

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)¶: Bases: sage.rings.padics.generic_nodes.CappedRelativeGeneric

class sage.rings.padics.generic_nodes.FixedModGeneric(base, prec, names, element_class)¶

Bases: sage.rings.padics.local_generic.LocalGeneric

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

class sage.rings.padics.generic_nodes.pAdicCappedRelativeFieldGeneric(base, p, prec, print_mode, names, element_class)¶: Bases: sage.rings.padics.generic_nodes.pAdicFieldGeneric, sage.rings.padics.generic_nodes.CappedRelativeFieldGeneric

class sage.rings.padics.generic_nodes.pAdicCappedRelativeRingGeneric(base, p, prec, print_mode, names, element_class)¶: Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.CappedRelativeRingGeneric

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

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

class sage.rings.padics.generic_nodes.pAdicFixedModRingGeneric(base, p, prec, print_mode, names, element_class)¶: Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.FixedModGeneric

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

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)¶

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

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

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$p$ -Adic Generic Nodes.¶