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