# $$p$$-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)
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)
class sage.rings.padics.generic_nodes.CappedRelativeGeneric(base, prec, names, element_class, category=None)
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)
class sage.rings.padics.generic_nodes.FixedModGeneric(base, prec, names, element_class, category=None)
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
sage: S = ZpCA(5, 15)
sage: S.is_fixed_mod()
False
sage: S(5^7,absprec=9)
5^7 + O(5^9)

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

Returns whether this $$p$$-adic ring uses a floating point precision model.

Elements in the floating point model are stored by giving a valuation and a unit part. Arithmetic is done where the unit part is truncated modulo a fixed power of the uniformizer, stored in the precision cap of the parent.

EXAMPLES:

sage: R = ZpFP(5,15)
sage: R.is_floating_point()
True
sage: R(5^7,absprec=9)
5^7
sage: S = ZpCR(5,15)
sage: S.is_floating_point()
False
sage: S(5^7,absprec=9)
5^7 + O(5^9)

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

class sage.rings.padics.generic_nodes.pAdicCappedAbsoluteRingGeneric(base, p, prec, print_mode, names, element_class, category=None)
class sage.rings.padics.generic_nodes.pAdicCappedRelativeFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)
class sage.rings.padics.generic_nodes.pAdicCappedRelativeRingGeneric(base, p, prec, print_mode, names, element_class, category=None)
class sage.rings.padics.generic_nodes.pAdicFieldBaseGeneric(p, prec, print_mode, names, element_class)
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(forbid_frac_field=False)

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).

INPUT:

EXAMPLES:

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


We can get a completion functor by forbidding the fraction field:

sage: c, L = K.construction(forbid_frac_field=True); L
Rational Field
sage: c
Completion[17, prec=8]
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)
class sage.rings.padics.generic_nodes.pAdicFixedModRingGeneric(base, p, prec, print_mode, names, element_class, category=None)
class sage.rings.padics.generic_nodes.pAdicFloatingPointFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)
class sage.rings.padics.generic_nodes.pAdicFloatingPointRingGeneric(base, p, prec, print_mode, names, element_class, category=None)
class sage.rings.padics.generic_nodes.pAdicLatticeGeneric(p, prec, print_mode, names, label=None)

An implementation of the $$p$$-adic rationals with lattice precision.

INPUT:

• $$p$$ – the underlying prime number
• prec – the precision
• subtype – either "cap" or "float", specifying the precision model used for tracking precision
• label – a string or None (default: None)
convert_multiple(*elts)

Convert a list of elements to this parent.

NOTE:

This function tries to be sharp on precision as much as possible. In particular, if the precision of the input elements are handled by a lattice, diffused digits of precision are preserved during the conversion.

EXAMPLES:

sage: R = ZpLC(2)
sage: x = R(1, 10); y = R(1, 5)
sage: x,y = x+y, x-y


Remark that the pair $$(x,y)$$ has diffused digits of precision:

sage: x
2 + O(2^5)
sage: y
O(2^5)
sage: x + y
2 + O(2^11)

sage: R.precision().diffused_digits([x,y])
6


As a consequence, if we convert x and y separately, we loose some precision:

sage: R2 = ZpLC(2, label='copy')
sage: x2 = R2(x); y2 = R2(y)
sage: x2
2 + O(2^5)
sage: y2
O(2^5)
sage: x2 + y2
2 + O(2^5)

sage: R2.precision().diffused_digits([x2,y2])
0


On the other hand, this issue disappears when we use multiple conversion:

sage: x2,y2 = R2.convert_multiple(x,y)
sage: x2 + y2
2 + O(2^11)

sage: R2.precision().diffused_digits([x2,y2])
6

is_lattice_prec()

Returns 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)
sage: S.is_lattice_prec()
True
sage: x = S(25, 8)
sage: x - x
O(5^30)

label()

Return the label of this parent.

NOTE:

Labels can be used to distinguish between parents with the same defining data.

They are useful in the lattice precision framework in order to limit the size of the lattice modeling the precision (which is roughly the number of elements having this parent).

Elements of a parent with some label do not coerce to a parent with a different label. However conversions are allowed.

EXAMPLES:

sage: R = ZpLC(5)
sage: R.label()  # no label by default

sage: R = ZpLC(5, label='mylabel')
sage: R.label()
'mylabel'


Labels are typically useful to isolate computations. For example, assume that we first want to do some calculations with matrices:

sage: R = ZpLC(5, label='matrices')
sage: M = random_matrix(R, 4, 4)
sage: d = M.determinant()


Now, if we want to do another unrelated computation, we can use a different label:

sage: R = ZpLC(5, label='polynomials')
sage: S.<x> = PolynomialRing(R)
sage: P = (x-1)*(x-2)*(x-3)*(x-4)*(x-5)


Without labels, the software would have modeled the precision on the matrices and on the polynomials using the same lattice (manipulating a lattice of higher dimension can have a significant impact on performance).

precision()

Return the lattice precision object attached to this parent.

EXAMPLES:

sage: R = ZpLC(5, label='precision')
sage: R.precision()
Precision lattice on 0 objects (label: precision)

sage: x = R(1, 10); y = R(1, 5)
sage: R.precision()
Precision lattice on 2 objects (label: precision)


sage.rings.padics.lattice_precision.PrecisionLattice

precision_cap()

Return the relative precision cap for this ring if it is finite. Otherwise return the absolute precision cap.

EXAMPLES:

sage: R = ZpLC(3)
sage: R.precision_cap()
20
sage: R.precision_cap_relative()
20

sage: R = ZpLC(3, prec=(infinity,20))
sage: R.precision_cap()
20
sage: R.precision_cap_relative()
+Infinity
sage: R.precision_cap_absolute()
20

precision_cap_absolute()

Return the absolute precision cap for this ring.

EXAMPLES:

sage: R = ZpLC(3)
sage: R.precision_cap_absolute()
40

sage: R = ZpLC(3, prec=(infinity,20))
sage: R.precision_cap_absolute()
20

precision_cap_relative()

Return the relative precision cap for this ring.

EXAMPLES:

sage: R = ZpLC(3)
sage: R.precision_cap_relative()
20

sage: R = ZpLC(3, prec=(infinity,20))
sage: R.precision_cap_relative()
+Infinity

class sage.rings.padics.generic_nodes.pAdicRingBaseGeneric(p, prec, print_mode, names, element_class)
construction(forbid_frac_field=False)

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).

INPUT:

• forbid_frac_field – ignored, for compatibility with other p-adic types.

EXAMPLES:

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

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