numthy · PadicLocalFieldElement

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

Worked example

The complete checked program Calculate trace and norm in a concrete quadratic finite field uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Calculate trace and norm in a concrete quadratic finite field."""

from noether.numthy import FiniteField


def main() -> None:
    field = FiniteField.extension(5, 2)
    generator = field.generator()
    assert generator**2 + field(4) * generator + field(2) == field.zero()
    assert generator.trace() == field(1)
    assert generator.norm() == field(2)
    print(f"{field}: Tr(a)={generator.trace()}, Nm(a)={generator.norm()}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/08_finite_field_extensions.py.

numthy.PadicLocalFieldElement

One normalized finite-precision value in a PadicLocalField.

PythonRust

class PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement · Direct Rust declaration

pub struct PadicLocalFieldElement { /* 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/PadicLocalFieldElement@1

Executable usage: validation/unit/infra/test_discovery.py:247 · validation/unit/infra/test_discovery.py:38 · validation/unit/infra/test_public_api_index.py:115 · validation/unit/numthy/test_padic_local_field_facade.py:15 · validation/unit/numthy/test_padic_local_field_facade.py:65 · validation/unit/numthy/test_padic_local_field_facade.py:83

numthy.PadicLocalFieldElement.__add__

Add another compatible value to this value.

PythonRust

(self, other: PadicLocalFieldElement) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement · Python projection

Python projection convenience over this Rust mathematical type

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/PadicLocalFieldElement@1

Executable usage: validation/unit/infra/test_discovery.py:211 · validation/unit/infra/test_public_api_index.py:293

numthy.PadicLocalFieldElement.__mul__

Multiply this value by another compatible value.

PythonRust

(self, other: PadicLocalFieldElement) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement · Python projection

Python projection convenience over this Rust mathematical type

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/PadicLocalFieldElement@1

Executable usage: validation/unit/infra/test_public_api_index.py:214

numthy.PadicLocalFieldElement.__neg__

Return the additive inverse of this value.

PythonRust

(self) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement · Python projection

Python projection convenience over this Rust mathematical type

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/PadicLocalFieldElement@1

numthy.PadicLocalFieldElement.__pow__

Raise this value to the supplied power.

PythonRust

(self, exponent: int, modulus: None=None) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement · Python projection

Python projection convenience over this Rust mathematical type

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/PadicLocalFieldElement@1

numthy.PadicLocalFieldElement.__sub__

Subtract another compatible value from this value.

PythonRust

(self, other: PadicLocalFieldElement) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement · Python projection

Python projection convenience over this Rust mathematical type

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/PadicLocalFieldElement@1

Executable usage: validation/unit/infra/test_discovery.py:212

numthy.PadicLocalFieldElement.characteristic_polynomial

Compute characteristic polynomial for this PadicLocalFieldElement.

Detailed contract

Characteristic polynomial over Q_p of multiplication by this value.

PythonRust

(self) -> Polynomial[PadicField]

noether::numthy::PadicLocalFieldElement::characteristic_polynomial · Direct Rust declaration

pub fn characteristic_polynomial( &self, ) -> Result<Polynomial<PadicField>, PadicLocalInvariantError>

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/PadicLocalFieldElement@1

Executable usage: validation/unit/linalg/test_acceptance_workload.py:79 · validation/unit/linalg/test_acceptance_workload.py:80 · validation/unit/linalg/test_acceptance_workload.py:98 · validation/unit/linalg/test_acceptance_workload.py:99 · validation/unit/linalg/test_exact_linear_algebra_facade.py:129 · validation/unit/linalg/test_exact_linear_algebra_facade.py:130

numthy.PadicLocalFieldElement.field

Return the field recorded by this PadicLocalFieldElement.

Detailed contract

Concrete local field presentation.

PythonRust

(self) -> PadicLocalField

noether::numthy::PadicLocalFieldElement::field · Direct Rust declaration

pub fn field(&self) -> &PadicLocalField

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/PadicLocalFieldElement@1

Executable usage: examples/08_finite_field_extensions.py:10 · examples/08_finite_field_extensions.py:11 · examples/08_finite_field_extensions.py:12 · examples/08_finite_field_extensions.py:13 · examples/08_finite_field_extensions.py:14 · examples/08_finite_field_extensions.py:3

numthy.PadicLocalFieldElement.from_json

Deserialize and revalidate a PadicLocalFieldElement from typed JSON.

Detailed contract

Deserialize and revalidate staggered coordinate contexts.

PythonRust

(cls, value: str) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement::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/PadicLocalFieldElement@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.PadicLocalFieldElement.inverse

Compute inverse for this PadicLocalFieldElement.

Detailed contract

Multiplicative inverse, preserving relative rather than absolute precision.

PythonRust

(self) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement::inverse · Direct Rust declaration

pub fn inverse(&self) -> Result<Self, UnitError>

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/PadicLocalFieldElement@1

Executable usage: examples/distributed/f2_linear_algebra.py:37 · examples/distributed/f2_linear_algebra.py:39 · validation/unit/commalg/test_context_tower_workload.py:121 · validation/unit/commalg/test_context_tower_workload.py:122 · validation/unit/commalg/test_context_tower_workload.py:75 · validation/unit/commalg/test_context_tower_workload.py:77

numthy.PadicLocalFieldElement.is_zero

Test whether this PadicLocalFieldElement is zero.

Detailed contract

Whether the value vanishes at its honest absolute precision.

PythonRust

(self) -> bool

noether::numthy::PadicLocalFieldElement::is_zero · Direct Rust declaration

pub fn is_zero(&self) -> bool

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/PadicLocalFieldElement@1

Executable usage: examples/17_weighted_power_series.py:16 · examples/17_weighted_power_series.py:17 · examples/39_kahler_differentials.py:20 · validation/unit/commalg/test_graded_commutative_facade.py:26 · validation/unit/commalg/test_kahler_differentials.py:113 · validation/unit/commalg/test_kahler_differentials.py:38

numthy.PadicLocalFieldElement.known_absolute_precision

Return the known absolute precision recorded by this PadicLocalFieldElement.

Detailed contract

Known absolute precision in powers of the distinguished uniformizer.

PythonRust

(self) -> int

noether::numthy::PadicLocalFieldElement::known_absolute_precision · Direct Rust declaration

pub const fn known_absolute_precision(&self) -> i64

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/PadicLocalFieldElement@1

Executable usage: validation/unit/numthy/test_acceptance_workload.py:106 · validation/unit/numthy/test_padic_local_field_facade.py:67 · validation/unit/numthy/test_padic_local_field_facade.py:96 · validation/unit/numthy/test_padic_ring_facade.py:68 · validation/unit/numthy/test_qadic_ring_facade.py:55 · validation/corpus/numthy/padic-newton-cube-root.py:23

numthy.PadicLocalFieldElement.minimal_polynomial

Compute minimal polynomial for this PadicLocalFieldElement.

Detailed contract

Minimal polynomial over Q_p when a full power basis is certified. A nonzero determinant for 1, a, ..., a^(n-1) proves that a generates the degree-n extension, in which case its minimal and characteristic polynomials agree. If that determinant vanishes at the available cap, this operation fails rather than infer a smaller degree from finite-precision zeros.

PythonRust

(self) -> Polynomial[PadicField]

noether::numthy::PadicLocalFieldElement::minimal_polynomial · Direct Rust declaration

pub fn minimal_polynomial( &self, ) -> Result<Polynomial<PadicField>, PadicLocalInvariantError>

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/PadicLocalFieldElement@1

Executable usage: validation/unit/infra/test_discovery.py:247 · validation/unit/numthy/test_number_field_facade.py:51 · validation/unit/numthy/test_padic_local_field_facade.py:80 · validation/unit/numthy/test_padic_local_field_facade.py:82 · validation/bench/magma_release.py:1460 · validation/bench/magma_release.py:1468

numthy.PadicLocalFieldElement.norm

Compute norm for this PadicLocalFieldElement.

Detailed contract

Field norm to Q_p at the largest uniformly justified coefficient cap.

PythonRust

(self) -> PadicElement

noether::numthy::PadicLocalFieldElement::norm · Direct Rust declaration

pub fn norm(&self) -> Result<PadicElement, PadicLocalInvariantError>

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/PadicLocalFieldElement@1

Executable usage: examples/06_shortest_lattice_vectors.py:15 · examples/08_finite_field_extensions.py:13 · examples/08_finite_field_extensions.py:14 · examples/08_finite_field_extensions.py:3 · examples/12_number_field_ideals.py:16 · examples/12_number_field_ideals.py:18

numthy.PadicLocalFieldElement.normalized_coordinates

Return the normalized coordinates recorded by this PadicLocalFieldElement.

Detailed contract

Staggered power-basis coordinates of the normalized integral unit. Zero has no normalized unit and therefore returns only absent coordinates. For a nonzero value, a missing coordinate means that its required relative cap is zero.

PythonRust

(self) -> list[QadicElement | None]

noether::numthy::PadicLocalFieldElement::normalized_coordinates · Direct Rust declaration

pub fn normalized_coordinates(&self) -> &[Option<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/PadicLocalFieldElement@1

numthy.PadicLocalFieldElement.pow

Compute pow for this PadicLocalFieldElement.

Detailed contract

Raise this value to a nonnegative word-sized power.

PythonRust

(self, exponent: int) -> PadicLocalFieldElement

noether::numthy::PadicLocalFieldElement::pow · Direct Rust declaration

pub fn pow(&self, exponent: u64) -> Self

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/PadicLocalFieldElement@1

Executable usage: examples/17_weighted_power_series.py:18 · validation/unit/commalg/test_polynomial_ideal_facade.py:368 · validation/unit/commalg/test_polynomial_ideal_facade.py:369 · validation/unit/commalg/test_polynomial_ideal_facade.py:370 · validation/unit/commalg/test_polynomial_ideal_facade.py:373 · validation/unit/commalg/test_polynomial_ideal_facade.py:381

numthy.PadicLocalFieldElement.relative_precision

Return the relative precision recorded by this PadicLocalFieldElement.

Detailed contract

Precision relative to the exact valuation of a nonzero value.

PythonRust

(self) -> int | None

noether::numthy::PadicLocalFieldElement::relative_precision · Direct Rust declaration

pub fn relative_precision(&self) -> Option<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/PadicLocalFieldElement@1

numthy.PadicLocalFieldElement.to_json

Serialize this PadicLocalFieldElement to its canonical typed JSON representation.

Detailed contract

Canonical typed JSON.

PythonRust

(self) -> str

noether::numthy::PadicLocalFieldElement::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/PadicLocalFieldElement@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.PadicLocalFieldElement.trace

Compute trace for this PadicLocalFieldElement.

Detailed contract

Field trace to Q_p at the largest uniformly justified coefficient cap.

PythonRust

(self) -> PadicElement

noether::numthy::PadicLocalFieldElement::trace · Direct Rust declaration

pub fn trace(&self) -> Result<PadicElement, PadicLocalInvariantError>

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/PadicLocalFieldElement@1

Executable usage: examples/08_finite_field_extensions.py:12 · examples/08_finite_field_extensions.py:14 · examples/08_finite_field_extensions.py:3 · validation/unit/commalg/test_acceptance_workload.py:41 · validation/unit/commalg/test_context_tower_workload.py:116 · validation/unit/commalg/test_context_tower_workload.py:96

numthy.PadicLocalFieldElement.valuation

Compute valuation for this PadicLocalFieldElement.

Detailed contract

Exact valuation, or a lower bound for a value zero at available precision.

PythonRust

(self) -> PadicValuation

noether::numthy::PadicLocalFieldElement::valuation · Direct Rust declaration

pub fn valuation(&self) -> PadicValuation

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/PadicLocalFieldElement@1

Executable usage: examples/31_local_fields_and_precision.py:37 · validation/unit/commalg/test_context_tower_workload.py:121 · validation/unit/commalg/test_context_tower_workload.py:122 · validation/unit/numthy/test_number_field_facade.py:244 · validation/unit/numthy/test_number_field_facade.py:245 · validation/unit/numthy/test_number_field_facade.py:255