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root · PrimeSubfieldEmbedding

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

Worked example

The complete checked program Apply and multiply canonical sparse exact operators uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Apply and multiply canonical sparse exact operators."""

from noether import Z
from noether.linalg import SparseMatrix, Vector


def main() -> None:
    operator = SparseMatrix.from_triplets(
        Z,
        (3, 3),
        [(0, 0, 2), (0, 2, -1), (1, 1, 3), (2, 0, 4)],
    )
    vector = Vector.from_values(Z, [5, 7, 11])
    image = operator.apply(vector)
    assert image.to_values() == [-1, 21, 20]
    assert (operator @ operator).nnz == 4
    print(f"A v = {image.to_values()}; nnz(A^2) = {(operator @ operator).nnz}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/05_sparse_exact_operators.py.

PrimeSubfieldEmbedding

The checked canonical embedding of a prime field into an extension field.

class PrimeSubfieldEmbedding

noether::PrimeSubfieldEmbedding · Direct Rust declaration

pub struct PrimeSubfieldEmbedding { /* 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/root/PrimeSubfieldEmbedding@1

Executable usage: validation/unit/test_coefficient_maps.py:14 · validation/unit/test_coefficient_maps.py:54

PrimeSubfieldEmbedding.__call__

Coerce the supplied input through this context or mathematical map.

(self, value: FiniteFieldElement) -> FiniteFieldElement

noether::root::PrimeSubfieldEmbedding · 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/root/PrimeSubfieldEmbedding@1

Executable usage: validation/unit/infra/test_discovery.py:57 · validation/unit/infra/test_public_api_index.py:155 · validation/unit/infra/test_public_api_index.py:169

PrimeSubfieldEmbedding.apply

Apply this PrimeSubfieldEmbedding to the supplied value.

(self, value: FiniteFieldElement) -> FiniteFieldElement

noether::root::PrimeSubfieldEmbedding::apply · Direct Rust declaration

fn apply( &self, value: &FiniteFieldElement, ) -> Result<FiniteFieldElement, Self::Error>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/root/PrimeSubfieldEmbedding@1

Executable usage: examples/05_sparse_exact_operators.py:16 · validation/unit/commalg/test_local_algebra.py:94 · validation/unit/commalg/test_local_algebra.py:97 · validation/unit/commalg/test_polynomial_facade.py:199 · validation/unit/commalg/test_quotient_facade.py:186 · validation/unit/groups/test_weight_characters.py:39

PrimeSubfieldEmbedding.from_json

Deserialize and revalidate a PrimeSubfieldEmbedding from typed JSON.

Detailed contract

Deserialize and exactly revalidate an embedding.

(cls, value: str) -> PrimeSubfieldEmbedding

noether::root::PrimeSubfieldEmbedding::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/root/PrimeSubfieldEmbedding@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

PrimeSubfieldEmbedding.source

Return the source recorded by this PrimeSubfieldEmbedding.

(self) -> FiniteField

noether::root::PrimeSubfieldEmbedding::source · Direct Rust declaration

fn source(&self) -> &Self::Source
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/root/PrimeSubfieldEmbedding@1

Executable usage: examples/09_finite_field_isomorphisms.py:11 · examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:9 · examples/30_modules_and_resolutions.py:34 · examples/30_modules_and_resolutions.py:37 · validation/unit/category/test_acceptance_workload.py:25

PrimeSubfieldEmbedding.target

Return the target recorded by this PrimeSubfieldEmbedding.

(self) -> FiniteField

noether::root::PrimeSubfieldEmbedding::target · Direct Rust declaration

fn target(&self) -> &Self::Target
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/root/PrimeSubfieldEmbedding@1

Executable usage: examples/09_finite_field_isomorphisms.py:10 · examples/09_finite_field_isomorphisms.py:11 · examples/30_modules_and_resolutions.py:33 · examples/30_modules_and_resolutions.py:38 · examples/31_local_fields_and_precision.py:27 · examples/36_affine_normalization.py:16

PrimeSubfieldEmbedding.to_json

Serialize this PrimeSubfieldEmbedding to its canonical typed JSON representation.

Detailed contract

Serialize both finite-field presentations.

(self) -> str

noether::root::PrimeSubfieldEmbedding::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/root/PrimeSubfieldEmbedding@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

PrimeSubfieldEmbedding.verify

Replay and verify the mathematical evidence carried by this PrimeSubfieldEmbedding.

Detailed contract

Verify that the source is prime and both characteristics agree.

(self) -> None

noether::root::PrimeSubfieldEmbedding::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), VerificationError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/root/PrimeSubfieldEmbedding@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12