fields

# Fields ## When to Use Use this skill when working on fields problems in abstract algebra. ## Decision Tree 1. **Is F a field?** - (F, +) is an abelian group with identity 0 - (F \ {0}, *) is an abelian group with identity 1 - Distributive law holds - `z3_solve.py prove "field_axioms"` 2. **Field Extensions** - E is extension of F if F is subfield of E - Degree [E:F] = dimension of E as F-vector space - `sympy_compute.py minpoly "alpha" --var x` for minimal polynomial 3. **Characteristic** - char(F) = smallest n > 0 where n*1 = 0, or 0 if none exists - char(F) is 0 or prime - For finite field: |F| = p^n where p = char(F) 4. **Algebraic Elements** - alpha is algebraic over F if it satisfies polynomial with coefficients in F - `sympy_compute.py solve "p(alpha) = 0"` for algebraic relations ## Tool Commands ### Z3_Field_Axioms ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "field_axioms" ``` ### Sympy_Minpoly ```bash uv run python -m runtime.harness scripts/sympy_compute.py minpoly "sqrt(2)" --var x ``` ### Sympy_Solve ```bash uv run python -m runtime.harness scripts/sympy_compute.py solve "x**2 - 2" --var x ``` ## Key Techniques *From indexed textbooks:* - [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 ...