Fields

When to Use

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

uv run python -m runtime.harness scripts/z3_solve.py prove "field_axioms"

Sympy_Minpoly

uv run python -m runtime.harness scripts/sympy_compute.py minpoly "sqrt(2)" --var x

Sympy_Solve

uv run python -m runtime.harness scripts/sympy_compute.py solve "x**2 - 2" --var x

Key Techniques

• [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.
• [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.
• [Catergories for the working mathematician] Geometric Functional Analysis and Its Applications. Lectures in Abstract Algebra II. Lectures in Abstract Algebra III.
• [Abstract Algebra] For p an odd prime, (Z/pZ) 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)*.
• [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.

Cognitive Tools Reference