A field F has (F,+) as an abelian group with identity 0, (F\{0},*) as an abelian group with identity 1, and the distributive law of multiplication over addition holds.

How do I compute the minimal polynomial of an algebraic element?

Use a tool like Sympy_Minpoly with the element and a variable, e.g., compute minpoly for sqrt(2) over Q. The command pattern is to call the relevant script (minpoly) on the element to obtain a polynomial p(x) in F[x] with p(alpha)=0.

How do I determine the size and char of a finite field?

The characteristic p is the smallest positive n with n*1 = 0 (or 0 if none). For a finite field, its size is |F| = p^n, where p = char(F) and n is the degree of the extension over its prime subfield.

fields

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SKILL.md

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Fields

When to Use

Use this skill when working on fields problems in abstract algebra.

Decision Tree

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"
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
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)
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

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 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

See .claude/skills/math-mode/SKILL.md for full tool documentation.

Source

git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/abstract-algebra/fields/SKILL.md

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Overview

This skill guides you through fields in abstract algebra with a practical decision-tree approach. It covers verifying field axioms, handling field extensions and characteristic, and identifying algebraic elements, using tool-assisted checks to illuminate structure and invariants.

How This Skill Works

You start by confirming the four field axioms, then move to field extensions, characteristic, and algebraic elements. At each step, the workflow recommends exact commands (Z3_Field_Axioms, Sympy_Minpoly, Sympy_Solve) to validate properties and compute minimal polynomials or solve relations, ensuring concrete, testable results.

When to Use It

Verifying that a candidate set with + and * satisfies field axioms (closure, inverses, distributivity, identity).
Analyzing a field extension E/F and computing the degree [E:F].
Determining the characteristic of a field and its implications for arithmetic (finite fields have size p^n).
Checking whether a given element alpha is algebraic over F and finding its minimal polynomial.
Computing minimal polynomials or solving equations over a field to understand algebraic relations.

Quick Start

Step 1: Identify whether the problem concerns field axioms, a field extension, characteristic, or algebraic elements.
Step 2: Use the decision-tree steps to select the relevant checks and computations.
Step 3: Run the appropriate tools (Z3_Field_Axioms, Sympy_Minpoly, Sympy_Solve) and record results.

Best Practices

Explicitly verify the four field axioms (additive/multiplicative identity, inverses, associativity, distributivity) before proceeding.
Follow the decision-tree steps to structure reasoning about field properties.
Use the provided tools: Z3_Field_Axioms to prove axioms, Sympy_Minpoly for minimal polynomials, and Sympy_Solve for solving equations.
For finite fields, keep track of characteristic p and size |F| = p^n.
Document results with clear citations to the corresponding axiom or theorem and note any assumptions.

Example Use Cases

Proving that the rational numbers Q form a field and using this as a baseline for comparison with rings Z/nZ.
Verifying whether Z/4Z is a field by analyzing whether every nonzero element has a multiplicative inverse.
Computing the minimal polynomial of sqrt(2) over Q using Sympy_Minpoly.
Determining the characteristic of a finite field and confirming its order |F| = p^n.
Calculating the degree [E:F] for a simple algebraic extension E = F(alpha) when alpha is algebraic over F.

Frequently Asked Questions

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