{"id":"fields","name":"fields","summary":"抽象代数における体の問題解決戦略","body":"# Fields\n\n## When to Use\n\nUse this skill when working on fields problems in abstract algebra.\n\n## Decision Tree\n\n\n1. **Is F a field?**\n   - (F, +) is an abelian group with identity 0\n   - (F \\ {0}, *) is an abelian group with identity 1\n   - Distributive law holds\n   - `z3_solve.py prove \"field_axioms\"`\n\n2. **Field Extensions**\n   - E is extension of F if F is subfield of E\n   - Degree [E:F] = dimension of E as F-vector space\n   - `sympy_compute.py minpoly \"alpha\" --var x` for minimal polynomial\n\n3. **Characteristic**\n   - char(F) = smallest n > 0 where n*1 = 0, or 0 if none exists\n   - char(F) is 0 or prime\n   - For finite field: |F| = p^n where p = char(F)\n\n4. **Algebraic Elements**\n   - alpha is algebraic over F if it satisfies polynomial with coefficients in F\n   - `sympy_compute.py solve \"p(alpha) = 0\"` for algebraic relations\n\n\n## Tool Commands\n\n### Z3_Field_Axioms\n```bash\nuv run python -m runtime.harness scripts/z3_solve.py prove \"field_axioms\"\n```\n\n### Sympy_Minpoly\n```bash\nuv run python -m runtime.harness scripts/sympy_compute.py minpoly \"sqrt(2)\" --var x\n```\n\n### Sympy_Solve\n```bash\nuv run python -m runtime.harness scripts/sympy_compute.py solve \"x**2 - 2\" --var x\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [Abstract Algebra] Write a computer program to add and multiply mod n, for any n given as input. The output of these operations should be the least residues of the sums and products of two integers. Also include the feature that if (a,n) = 1, an integer c between 1 and n — 1 such that a-c = | may be printed on request.\n- [Abstract Algebra] Reading the above equation mod4\\(that is, considering this equation in the quotient ring Z/4Z), we must have {2} =2[9}=[9} ons ( io ‘| where the | he? Checking the few saad shows that we must take the 0 each time. Introduction to Rings Another ideal in RG is {}-\"_, agi | a € R}, i.\n- [Catergories for the working mathematician] Geometric Functional Analysis and Its Applications. Lectures in Abstract Algebra II. Lectures in Abstract Algebra III.\n- [Abstract Algebra] For p an odd prime, (Z/p*Z)* is an abelian group of order p* ‘(p — 1). Sylow p-subgroup of this group is cyclic. The map Z/p°Z > Z/pZ defined by at+(p*) a+t+(p) is a ring homomorphism (reduction mod p) which gives a surjective group homo- morphism from (Z/p%Z)* onto (Z/pZ)*.\n- [A Classical Introduction to Modern Number Theory (Graduate] Graduate Texts in Mathematics 84 Editorial Board s. Ribet Springer Science+Business Media, LLC 2 3 TAKEUTtlZARING. Introduction to Axiomatic Set Theory.\n\n## Cognitive Tools Reference\n\nSee `.claude/skills/math-mode/SKILL.md` for full tool documentation.","author":"@parcadei","ownerProfile":null,"authorContacts":null,"sourceUrl":"https://github.com/parcadei/Continuous-Claude-v3/tree/main/.claude/skills/math/abstract-algebra/fields","license":"MIT","category":"writing","lang":"en","tokens":749,"stars":0,"calls30d":2,"claimed":false,"visibility":"public","origin":"crawler","version":"0.1.0","createdAt":"2026-08-22","updatedAt":"2026-08-22","files":[],"requires":{"mcp":[],"tools":["Bash","Read"]},"safety":{"flags":[],"scannedAt":"2026-08-22","hasScripts":false,"networkEndpoints":[]}}