[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"wRFucxQJWl":3},"# ClassFieldTheory (Lean 4.35.0-rc2)\n\n[![Lean](https://github.com/n-yamaguchi-0729/ClassFieldTheory/actions/workflows/lean.yml/badge.svg)](https://github.com/n-yamaguchi-0729/ClassFieldTheory/actions/workflows/lean.yml)\n[![License](https://img.shields.io/badge/license-Apache--2.0-blue.svg)](LICENSE)\n\nDeveloped as part of [NYLL](https://github.com/n-yamaguchi-0729/NYLL), this Lean library treats class field theory for\nnonarchimedean local fields, number fields and function fields over finite fields.\nThe [mathematical guide](https://n-yamaguchi-0729.github.io/NYLL/libraries/class-field-theory/) presents reciprocity, existence and the other selected results\nwith their mathematical hypotheses and links to the formal proofs.\n\n## Main results\n\n\u003C!-- nyll-publication:cft-main-results:begin -->\n### Local class field theory\n\nLet K be a nonarchimedean local field.\n\n| Result | Mathematical content |\n| --- | --- |\n| [Finite abelian reciprocity](Nyll/ClassFieldTheory/Theorems/LocalClassFieldTheory/FiniteAbelianLocalReciprocity.lean#L33) | For each finite abelian extension L/K, a continuous surjective Artin map Kˣ → Gal(L/K) has kernel N(Lˣ). |\n| [Existence and classification](Nyll/ClassFieldTheory/Theorems/LocalClassFieldTheory/FiniteAbelianLocalExistenceOrderIso.lean#L33) | Finite abelian extensions in a fixed separable closure correspond, with reversed inclusion, to open finite-index subgroups of Kˣ via norms. |\n\n### Global class field theory: number fields\n\nLet K be a number field.\n\n| Result | Mathematical content |\n| --- | --- |\n| [Finite abelian reciprocity](Nyll/ClassFieldTheory/Theorems/GlobalClassFieldTheory/FiniteAbelianGlobalReciprocity.lean#L31) | Each finite abelian extension L/K admits a modulus and a surjective ray-class Artin map to Gal(L/K), sending prime classes away from the modulus to arithmetic Frobenius. |\n| [Class-field existence](Nyll/ClassFieldTheory/Theorems/ConductorsAndRayClassFields/RayClassSubgroupExistence.lean#L27) | Every subgroup of a ray class group is the kernel of the Frobenius-normalized Artin map of a finite abelian extension. |\n\n### Global class field theory: function fields\n\nLet K be a function field in one variable over a finite field k.\n\n| Result | Mathematical content |\n| --- | --- |\n| [Finite abelian reciprocity](Nyll/ClassFieldTheory/Theorems/FunctionField/FiniteReciprocity.lean#L66) | For each finite abelian extension L/K, the Artin map identifies C_K / N(C_L) with Gal(L/K), where C denotes the idèle class group. |\n| [Class-field existence and uniqueness](Nyll/ClassFieldTheory/Theorems/FunctionField/ClassField.lean#L60) | Every open finite-index subgroup of C_K in its usual topology is the norm subgroup of a unique finite abelian extension in a fixed separable closure. |\n\u003C!-- nyll-publication:cft-main-results:end -->\n\n## Public API\n\nUse [leanprover/lean4:v4.35.0-rc2](lean-toolchain) and the Mathlib revision pinned in\n[lake-manifest.json](lake-manifest.json). Add this dependency to your project’s\nlakefile.toml:\n\n```toml\n[[require]]\nname = \"Nyll\"\ngit = \"https://github.com/n-yamaguchi-0729/ClassFieldTheory.git\"\nrev = \"main\"\n```\n\nRun `lake update Nyll`, then commit your generated lake-manifest.json to retain\nthe resolved revisions. Import the library in a Lean file:\n\n```lean\nimport Nyll.ClassFieldTheory.All\n```\n\nThe standalone distributions share the package name Nyll. Choose one distribution;\nfor several libraries together, use [NYLL](https://github.com/n-yamaguchi-0729/NYLL).\n\nTo build a checkout of this repository, run from its root:\n\n```sh\nlake exe cache get\nlake --wfail build\n```\n\n## Verification\n\n[CI](.github/workflows/lean.yml) checks the import closure, builds with\nwarnings as errors, audits declarations, and verifies proofs with NanoDa\nand the Lean kernel. The [public API contract](.github/verification/main-declarations.json)\nchecks the named declarations and their source modules.\n\n## Authorship\n\nAll original code in this library is written by OpenAI Codex.\n\n[Naganori Yamaguchi](https://github.com/n-yamaguchi-0729) sets the mathematical\ngoals and scope, directs development, and maintains the library.\n\n## License\n\n\u003C!-- nyll-publication:credits:begin -->\nApache License 2.0. See [LICENSE](LICENSE).\n\nIncluded adaptations from [pi1](https://github.com/chrisflav/pi1) and [Anthropic’s Fermat’s Last Theorem project](https://github.com/anthropics/fermats-last-theorem) retain their source-specific copyrights and authors.\nThe included-file inventories and preserved upstream notices are in [NOTICE](NOTICE).\n\u003C!-- nyll-publication:credits:end -->\n",1793400797699]