numthy · PadicLocalField¶

29 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example¶

The complete checked program Use a ramified local field, its maximal order, and explicit precision loss uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Use a ramified local field, its maximal order, and explicit precision loss."""

from noether.commalg import PolynomialRing
from noether.numthy import FiniteField, PadicLocalField, PadicLocalPrecision, QadicRing


def main() -> None:
    residue = FiniteField.extension(3, 2)
    coefficients = QadicRing(residue, absolute_precision=4)
    polynomials = PolynomialRing(coefficients, ["T"])
    t = polynomials.generator(0)
    field = PadicLocalField(
        coefficients,
        t**2 + 3,
        PadicLocalPrecision(7),
        uniformizer_name="pi",
    )

    pi = field.uniformizer()
    order = field.maximal_order()
    frobenius = field.coefficient_frobenius_conjugate(1)
    frobenius.verify()
    assert order.is_maximal()
    assert pi in order
    assert frobenius(field.one() + pi).field == frobenius.target

    try:
        field.precision_restriction(PadicLocalPrecision(8))
    except ValueError as error:
        precision_failure = type(error).__name__
    else:
        raise AssertionError("increasing fixed precision must fail closed")

    print(
        f"degree={field.degree}, discriminant valuation={order.discriminant_valuation()}, "
        f"Frobenius exponent={frobenius.exponent}, precision failure={precision_failure}"
    )


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/31_local_fields_and_precision.py.

`numthy.PadicLocalField`¶

A finite extension of Q_p in explicit unramified-plus-Eisenstein form.

PythonRust

PadicLocalField(coefficient_ring: QadicRing, eisenstein_polynomial: Polynomial[QadicRing], precision: PadicLocalPrecision, *, uniformizer_name: str | None=None)

noether::numthy::PadicLocalField · Direct Rust declaration

pub struct PadicLocalField { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/31_local_fields_and_precision.py:14 · examples/31_local_fields_and_precision.py:6 · validation/unit/infra/test_public_api_index.py:184 · validation/unit/numthy/test_padic_local_field_facade.py:13 · validation/unit/numthy/test_padic_local_field_facade.py:151 · validation/unit/numthy/test_padic_local_field_facade.py:24

`numthy.PadicLocalField.coefficient_field`¶

Return the coefficient field recorded by this PadicLocalField.

Detailed contract

Maximal unramified coefficient field in the declared presentation.

PythonRust

(self) -> QadicField

noether::numthy::PadicLocalField::coefficient_field · Direct Rust declaration

pub fn coefficient_field(&self) -> &QadicField

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/linalg/test_semilinear_facade.py:53 · validation/unit/numthy/test_padic_local_field_facade.py:49 · validation/unit/numthy/test_padic_local_field_facade.py:78 · validation/unit/numthy/test_qadic_field_facade.py:145 · validation/unit/numthy/test_qadic_field_facade.py:146 · validation/unit/numthy/test_qadic_field_facade.py:31

`numthy.PadicLocalField.coefficient_frobenius_conjugate`¶

Compute coefficient frobenius conjugate for this PadicLocalField.

Detailed contract

Conjugate this presentation by a lifted coefficient Frobenius power.

PythonRust

(self, exponent: int) -> PadicLocalFieldCoefficientFrobenius

noether::numthy::PadicLocalField::coefficient_frobenius_conjugate · Direct Rust declaration

pub fn coefficient_frobenius_conjugate( &self, exponent: u64, ) -> Result<PadicLocalFieldCoefficientFrobenius, PadicLocalCoefficientFrobeniusError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/31_local_fields_and_precision.py:23 · validation/unit/numthy/test_padic_local_field_facade.py:106 · validation/unit/numthy/test_padic_local_field_facade.py:107 · validation/unit/numthy/test_padic_local_field_facade.py:157

`numthy.PadicLocalField.coefficient_ring`¶

Return the coefficient ring recorded by this PadicLocalField.

Detailed contract

Original fixed-cap Witt coefficient ring used by the presentation.

PythonRust

(self) -> QadicRing

noether::numthy::PadicLocalField::coefficient_ring · Direct Rust declaration

pub fn coefficient_ring(&self) -> &QadicRing

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/commalg/test_affine_normalization_facade.py:115 · validation/unit/commalg/test_affine_normalization_facade.py:117 · validation/unit/commalg/test_affine_normalization_facade.py:119 · validation/unit/commalg/test_base_change.py:31 · validation/unit/commalg/test_base_change.py:53 · validation/unit/commalg/test_construction_algebras.py:19

`numthy.PadicLocalField.coordinate_precisions`¶

Return the coordinate precisions recorded by this PadicLocalField.

Detailed contract

Staggered p-adic caps for 1, pi, ..., pi^(e-1).

PythonRust

(self) -> list[int]

noether::numthy::PadicLocalField::coordinate_precisions · Direct Rust declaration

pub fn coordinate_precisions(&self) -> &[u64]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_padic_local_field_facade.py:47

`numthy.PadicLocalField.defining_coefficients`¶

Return the defining coefficients recorded by this PadicLocalField.

Detailed contract

Ascending coefficients of the defining Eisenstein polynomial.

PythonRust

(self) -> list[QadicElement]

noether::numthy::PadicLocalField::defining_coefficients · Direct Rust declaration

pub fn defining_coefficients(&self) -> &[QadicElement]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

`numthy.PadicLocalField.defining_polynomial`¶

Return the defining polynomial recorded by this PadicLocalField.

Detailed contract

Monic Eisenstein polynomial defining the distinguished uniformizer.

PythonRust

(self) -> Polynomial[QadicRing]

noether::numthy::PadicLocalField::defining_polynomial · Direct Rust declaration

pub fn defining_polynomial(&self) -> &Polynomial<QadicRing>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_number_field_facade.py:25 · validation/unit/numthy/test_padic_local_field_facade.py:51 · validation/corpus/numthy/degree-six-ring-of-integers.py:15 · validation/corpus/numthy/degree-six-ring-of-integers.py:16

`numthy.PadicLocalField.degree`¶

Return the degree recorded by this PadicLocalField.

Detailed contract

Absolute degree over Q_p.

PythonRust

(self) -> int

noether::numthy::PadicLocalField::degree · Direct Rust declaration

pub fn degree(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/14_weighted_polynomials.py:16 · examples/14_weighted_polynomials.py:18 · examples/15_groebner_twisted_cubic.py:15 · examples/15_groebner_twisted_cubic.py:28 · examples/31_local_fields_and_precision.py:37 · examples/33_simplicial_cohomology_operations.py:26

`numthy.PadicLocalField.element_from_padic`¶

Compute element from padic for this PadicLocalField.

Detailed contract

Embed a scalar from the exact base Q_p context.

PythonRust

(self, value: PadicElement) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::element_from_padic · Direct Rust declaration

pub fn element_from_padic( &self, value: &PadicElement, ) -> Result<PadicLocalFieldElement, PadicLocalFieldError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

`numthy.PadicLocalField.element_from_qadic`¶

Compute element from qadic for this PadicLocalField.

Detailed contract

Embed a scalar from the exact maximal unramified coefficient field. The scalar’s p-adic valuation is scaled by the ramification index. A negative scalar can consequently have relative precision larger than the field’s maximum absolute cap; its normalized unit coordinates keep precisely those extra digits rather than rounding them away.

PythonRust

(self, value: QadicElement) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::element_from_qadic · Direct Rust declaration

pub fn element_from_qadic( &self, value: &QadicElement, ) -> Result<PadicLocalFieldElement, PadicLocalFieldError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_padic_local_field_facade.py:78

`numthy.PadicLocalField.equation_order`¶

Compute equation order for this PadicLocalField.

Detailed contract

The order generated by the declared unramified generator and uniformizer. Every currently admitted local field is Eisenstein-presented, so this equation order is the maximal order by construction.

PythonRust

(self) -> PadicLocalFieldOrder

noether::numthy::PadicLocalField::equation_order · Direct Rust declaration

pub fn equation_order( &self, ) -> Result<PadicLocalFieldOrder, PadicLocalOrderError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_number_field_facade.py:127 · validation/unit/numthy/test_number_field_facade.py:141 · validation/unit/numthy/test_number_field_facade.py:150 · validation/unit/numthy/test_number_field_facade.py:156 · validation/unit/numthy/test_number_field_facade.py:165 · validation/unit/numthy/test_number_field_facade.py:95

`numthy.PadicLocalField.from_json`¶

Deserialize and revalidate a PadicLocalField from typed JSON.

Detailed contract

Deserialize and revalidate an Eisenstein presentation.

PythonRust

(cls, value: str) -> PadicLocalField

noether::numthy::PadicLocalField::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

`numthy.PadicLocalField.integral_element`¶

Compute integral element for this PadicLocalField.

Detailed contract

Construct an integral element from ascending coefficients in the original presentation ring, discarding only digits outside each staggered coordinate cap.

PythonRust

(self, coefficients: list[QadicElement]) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::integral_element · Direct Rust declaration

pub fn integral_element( &self, coefficients: &[QadicElement], ) -> Result<PadicLocalFieldElement, PadicLocalFieldError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_padic_local_field_facade.py:185 · validation/unit/numthy/test_padic_local_field_facade.py:241

`numthy.PadicLocalField.local_precision`¶

Return the local precision recorded by this PadicLocalField.

Detailed contract

Absolute uniformizer precision.

PythonRust

(self) -> PadicLocalPrecision

noether::numthy::PadicLocalField::local_precision · Direct Rust declaration

pub fn local_precision(&self) -> PadicLocalPrecision

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_padic_local_field_facade.py:50 · validation/unit/numthy/test_padic_local_field_facade.py:94

`numthy.PadicLocalField.maximal_order`¶

Compute and certify the maximal order of this finite p-adic local field.

Detailed contract

The maximal order O_K = W(k)[pi] in the admitted Eisenstein presentation.

PythonRust

(self) -> PadicLocalFieldOrder

noether::numthy::PadicLocalField::maximal_order · Direct Rust declaration

pub fn maximal_order( &self, ) -> Result<PadicLocalFieldOrder, PadicLocalOrderError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/12_number_field_ideals.py:13 · examples/31_local_fields_and_precision.py:22 · validation/unit/infra/test_public_api_index.py:165 · validation/unit/infra/test_public_api_index.py:184 · validation/unit/numthy/test_acceptance_workload.py:50 · validation/unit/numthy/test_number_field_facade.py:142

`numthy.PadicLocalField.one`¶

Compute one for this PadicLocalField.

Detailed contract

Multiplicative identity at the declared staggered precision.

PythonRust

(self) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::one · Direct Rust declaration

pub fn one(&self) -> PadicLocalFieldElement
fn one(&self) -> Self::Element

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/02_ball_arithmetic.py:13 · examples/11_qadic_witt_vectors.py:11 · examples/24_homology_and_persistence.py:3 · examples/31_local_fields_and_precision.py:27 · validation/unit/combinatorics/test_poset.py:3 · validation/unit/commalg/test_acceptance_workload.py:37

`numthy.PadicLocalField.order`¶

Compute order for this PadicLocalField.

Detailed contract

Validate an arbitrary full Z_p-basis as a local order.

PythonRust

(self, basis: list[PadicLocalFieldElement]) -> PadicLocalFieldOrder

noether::numthy::PadicLocalField::order · Direct Rust declaration

pub fn order( &self, basis: &[PadicLocalFieldElement], ) -> Result<PadicLocalFieldOrder, PadicLocalOrderError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/11_qadic_witt_vectors.py:14 · examples/12_number_field_ideals.py:13 · examples/12_number_field_ideals.py:14 · examples/12_number_field_ideals.py:17 · examples/15_groebner_twisted_cubic.py:18 · examples/19_permutation_groups.py:13

`numthy.PadicLocalField.precision_restriction`¶

Compute precision restriction for this PadicLocalField.

Detailed contract

Construct a first-class restriction to a coarser uniformizer cap.

PythonRust

(self, target_precision: PadicLocalPrecision) -> PadicLocalFieldRestriction

noether::numthy::PadicLocalField::precision_restriction · Direct Rust declaration

pub fn precision_restriction( &self, target_precision: PadicLocalPrecision, ) -> Result<PadicLocalFieldRestriction, PadicLocalFieldRestrictionError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/31_local_fields_and_precision.py:30 · validation/unit/commalg/test_power_series_facade.py:319 · validation/unit/homalg/test_padic_homology_facade.py:204 · validation/unit/numthy/test_padic_field_facade.py:58 · validation/unit/numthy/test_padic_field_facade.py:78 · validation/unit/numthy/test_padic_local_field_facade.py:101

`numthy.PadicLocalField.prime`¶

Return the prime recorded by this PadicLocalField.

Detailed contract

Residue characteristic p.

PythonRust

(self) -> int

noether::numthy::PadicLocalField::prime · Direct Rust declaration

pub fn prime(&self) -> &Integer

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/07_integer_factorization.py:12 · examples/12_number_field_ideals.py:16 · examples/12_number_field_ideals.py:19 · examples/18_finite_field_schemes.py:10 · examples/24_homology_and_persistence.py:22 · examples/33_simplicial_cohomology_operations.py:24

`numthy.PadicLocalField.ramification_index`¶

Return the ramification index recorded by this PadicLocalField.

Detailed contract

Ramification index over the maximal unramified coefficient field.

PythonRust

(self) -> int

noether::numthy::PadicLocalField::ramification_index · Direct Rust declaration

pub fn ramification_index(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_acceptance_workload.py:59 · validation/unit/numthy/test_acceptance_workload.py:68 · validation/unit/numthy/test_acceptance_workload.py:69 · validation/unit/numthy/test_acceptance_workload.py:72 · validation/unit/numthy/test_number_field_facade.py:213 · validation/unit/numthy/test_number_field_facade.py:239

`numthy.PadicLocalField.residue`¶

Compute residue for this PadicLocalField.

Detailed contract

Reduce an integral value to the explicit finite residue field. None means that the value has negative valuation, or that a zero is known only at nonpositive absolute precision and therefore has no determined residue.

PythonRust

(self, value: PadicLocalFieldElement) -> FiniteFieldElement | None

noether::numthy::PadicLocalField::residue · Direct Rust declaration

pub fn residue( &self, value: &PadicLocalFieldElement, ) -> Result<Option<FiniteFieldElement>, ContextMismatchError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/11_qadic_witt_vectors.py:11 · examples/11_qadic_witt_vectors.py:12 · examples/11_qadic_witt_vectors.py:13 · examples/11_qadic_witt_vectors.py:15 · examples/13_chinese_remainder_theorem.py:3 · examples/31_local_fields_and_precision.py:10

`numthy.PadicLocalField.residue_degree`¶

Return the residue degree recorded by this PadicLocalField.

Detailed contract

Residue degree over Q_p.

PythonRust

(self) -> int

noether::numthy::PadicLocalField::residue_degree · Direct Rust declaration

pub fn residue_degree(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/numthy/test_acceptance_workload.py:59 · validation/unit/numthy/test_acceptance_workload.py:68 · validation/unit/numthy/test_acceptance_workload.py:69 · validation/unit/numthy/test_acceptance_workload.py:72 · validation/unit/numthy/test_number_field_facade.py:213 · validation/unit/numthy/test_number_field_facade.py:234

`numthy.PadicLocalField.residue_field`¶

Return the residue field recorded by this PadicLocalField.

Detailed contract

Explicit finite residue field.

PythonRust

(self) -> FiniteField

noether::numthy::PadicLocalField::residue_field · Direct Rust declaration

pub fn residue_field(&self) -> &FiniteField

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/10_padic_hensel_lifting.py:14 · examples/11_qadic_witt_vectors.py:10 · examples/11_qadic_witt_vectors.py:11 · examples/11_qadic_witt_vectors.py:14 · examples/11_qadic_witt_vectors.py:9 · validation/unit/commalg/test_acceptance_workload.py:25

`numthy.PadicLocalField.restrict`¶

Compute restrict for this PadicLocalField.

Detailed contract

Explicitly discard high uniformizer digits into a distinct field context.

PythonRust

(self, value: PadicLocalFieldElement, target_precision: PadicLocalPrecision) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::restrict · Direct Rust declaration

pub fn restrict( &self, value: &PadicLocalFieldElement, target_precision: PadicLocalPrecision, ) -> Result<PadicLocalFieldElement, PadicLocalFieldRestrictionError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/commalg/test_local_algebra.py:93 · validation/unit/groups/test_weight_characters.py:48 · validation/unit/numthy/test_padic_local_field_facade.py:100 · validation/unit/numthy/test_padic_ring_facade.py:62 · validation/unit/numthy/test_qadic_ring_facade.py:48

`numthy.PadicLocalField.teichmuller`¶

Compute teichmuller for this PadicLocalField.

Detailed contract

Lift a residue-field value through the maximal unramified coefficient ring.

PythonRust

(self, value: FiniteFieldElement) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::teichmuller · Direct Rust declaration

pub fn teichmuller( &self, value: &FiniteFieldElement, ) -> Result<PadicLocalFieldElement, ContextMismatchError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/11_qadic_witt_vectors.py:12 · validation/unit/commalg/test_acceptance_workload.py:33 · validation/unit/commalg/test_acceptance_workload.py:38 · validation/unit/commalg/test_context_tower_workload.py:101 · validation/unit/commalg/test_context_tower_workload.py:113 · validation/unit/infra/test_public_api_index.py:164

`numthy.PadicLocalField.to_json`¶

Serialize this PadicLocalField to its canonical typed JSON representation.

Detailed contract

Canonical typed JSON preserving display metadata as well as identity.

PythonRust

(self) -> str

noether::numthy::PadicLocalField::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

`numthy.PadicLocalField.uniformizer`¶

Compute uniformizer for this PadicLocalField.

Detailed contract

Distinguished root of the defining Eisenstein polynomial.

PythonRust

(self) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::uniformizer · Direct Rust declaration

pub fn uniformizer(&self) -> PadicLocalFieldElement
fn uniformizer(&self) -> Self::Element

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/31_local_fields_and_precision.py:21 · validation/unit/linalg/test_acceptance_workload.py:100 · validation/unit/linalg/test_acceptance_workload.py:101 · validation/unit/linalg/test_acceptance_workload.py:65 · validation/unit/linalg/test_acceptance_workload.py:68 · validation/unit/linalg/test_acceptance_workload.py:70

`numthy.PadicLocalField.uniformizer_name`¶

Return the uniformizer name recorded by this PadicLocalField.

Detailed contract

Bounded display name of the distinguished uniformizer.

PythonRust

(self) -> str

noether::numthy::PadicLocalField::uniformizer_name · Direct Rust declaration

pub fn uniformizer_name(&self) -> &str

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/31_local_fields_and_precision.py:18 · validation/unit/numthy/test_padic_local_field_facade.py:35 · validation/unit/numthy/test_padic_local_field_facade.py:48

`numthy.PadicLocalField.zero`¶

Compute zero for this PadicLocalField.

Detailed contract

Additive identity at the declared staggered precision.

PythonRust

(self) -> PadicLocalFieldElement

noether::numthy::PadicLocalField::zero · Direct Rust declaration

pub fn zero(&self) -> PadicLocalFieldElement
fn zero(&self) -> Self::Element

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core
GIL	attached / bounded
Semantic ID	noether/numthy/PadicLocalField@1

Executable usage: examples/08_finite_field_extensions.py:11 · examples/10_padic_hensel_lifting.py:15 · examples/40_exterior_algebra.py:26 · examples/distributed/f2_linear_algebra.py:28 · examples/distributed/f2_linear_algebra.py:29 · examples/distributed/f2_linear_algebra.py:30

numthy · PadicLocalField¶

Worked example¶

numthy.PadicLocalField¶

numthy.PadicLocalField.coefficient_field¶

numthy.PadicLocalField.coefficient_frobenius_conjugate¶

numthy.PadicLocalField.coefficient_ring¶

numthy.PadicLocalField.coordinate_precisions¶

numthy.PadicLocalField.defining_coefficients¶

numthy.PadicLocalField.defining_polynomial¶

numthy.PadicLocalField.degree¶

numthy.PadicLocalField.element_from_padic¶

numthy.PadicLocalField.element_from_qadic¶

numthy.PadicLocalField.equation_order¶

numthy.PadicLocalField.from_json¶

numthy.PadicLocalField.integral_element¶

numthy.PadicLocalField.local_precision¶

numthy.PadicLocalField.maximal_order¶

numthy.PadicLocalField.one¶

numthy.PadicLocalField.order¶

numthy.PadicLocalField.precision_restriction¶

numthy.PadicLocalField.prime¶

numthy.PadicLocalField.ramification_index¶

numthy.PadicLocalField.residue¶

numthy.PadicLocalField.residue_degree¶

numthy.PadicLocalField.residue_field¶

numthy.PadicLocalField.restrict¶

numthy.PadicLocalField.teichmuller¶

numthy.PadicLocalField.to_json¶

numthy.PadicLocalField.uniformizer¶

numthy.PadicLocalField.uniformizer_name¶

numthy.PadicLocalField.zero¶

`numthy.PadicLocalField`¶

`numthy.PadicLocalField.coefficient_field`¶

`numthy.PadicLocalField.coefficient_frobenius_conjugate`¶

`numthy.PadicLocalField.coefficient_ring`¶

`numthy.PadicLocalField.coordinate_precisions`¶

`numthy.PadicLocalField.defining_coefficients`¶

`numthy.PadicLocalField.defining_polynomial`¶

`numthy.PadicLocalField.degree`¶

`numthy.PadicLocalField.element_from_padic`¶

`numthy.PadicLocalField.element_from_qadic`¶

`numthy.PadicLocalField.equation_order`¶

`numthy.PadicLocalField.from_json`¶

`numthy.PadicLocalField.integral_element`¶

`numthy.PadicLocalField.local_precision`¶

`numthy.PadicLocalField.maximal_order`¶

`numthy.PadicLocalField.one`¶

`numthy.PadicLocalField.order`¶

`numthy.PadicLocalField.precision_restriction`¶

`numthy.PadicLocalField.prime`¶

`numthy.PadicLocalField.ramification_index`¶

`numthy.PadicLocalField.residue`¶

`numthy.PadicLocalField.residue_degree`¶

`numthy.PadicLocalField.residue_field`¶

`numthy.PadicLocalField.restrict`¶

`numthy.PadicLocalField.teichmuller`¶

`numthy.PadicLocalField.to_json`¶

`numthy.PadicLocalField.uniformizer`¶

`numthy.PadicLocalField.uniformizer_name`¶

`numthy.PadicLocalField.zero`¶