fields

Name: fields
Author: parcadei

3.9kstars297forks

How to install fields?

parcadei/continuous-claude-v3/fields

$npx -y skills add parcadei/Continuous-Claude-v3 --skill fields

Installs into the current project.

Summary

Problem-solving strategies for fields in abstract algebra

Files · 1

View on GitHub

SKILL.md

1	# Fields
2
3	## When to Use
4
5	Use this skill when working on fields problems in abstract algebra.
6
7	## Decision Tree
8
9
10	1. Is F a field?
11	- (F, +) is an abelian group with identity 0
12	- (F \ {0}, *) is an abelian group with identity 1
13	- Distributive law holds
14	- `z3_solve.py prove "field_axioms"`
15
16	2. Field Extensions
17	- E is extension of F if F is subfield of E
18	- Degree [E:F] = dimension of E as F-vector space
19	- `sympy_compute.py minpoly "alpha" --var x` for minimal polynomial
20
21	3. Characteristic
22	- char(F) = smallest n > 0 where n*1 = 0, or 0 if none exists
23	- char(F) is 0 or prime
24	- For finite field: \|F\| = p^n where p = char(F)
25
26	4. Algebraic Elements
27	- alpha is algebraic over F if it satisfies polynomial with coefficients in F
28	- `sympy_compute.py solve "p(alpha) = 0"` for algebraic relations
29
30
31	## Tool Commands
32
33	### Z3_Field_Axioms
34	```bash
35	uv run python -m runtime.harness scripts/z3_solve.py prove "field_axioms"
36	```
37
38	### Sympy_Minpoly
39	```bash
40	uv run python -m runtime.harness scripts/sympy_compute.py minpoly "sqrt(2)" --var x
41	```
42
43	### Sympy_Solve
44	```bash
45	uv run python -m runtime.harness scripts/sympy_compute.py solve "x**2 - 2" --var x
46	```
47
48	## Key Techniques
49
50	From indexed textbooks:
51
52	- [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.
53	- [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.
54	- [Catergories for the working mathematician] Geometric Functional Analysis and Its Applications. Lectures in Abstract Algebra II. Lectures in Abstract Algebra III.
55	- [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)*.
56	- [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.
57
58	## Cognitive Tools Reference
59
60	See `.claude/skills/math-mode/SKILL.md` for full tool documentation.

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fields

3.9kstars297forks

View on GitHub

How to install fields?

parcadei/continuous-claude-v3/fields

$npx -y skills add parcadei/Continuous-Claude-v3 --skill fields

Installs into the current project.

Summary

Problem-solving strategies for fields in abstract algebra

Files · 1

View on GitHub

SKILL.md

1	# Fields
2
3	## When to Use
4
5	Use this skill when working on fields problems in abstract algebra.
6
7	## Decision Tree
8
9
10	1. Is F a field?
11	- (F, +) is an abelian group with identity 0
12	- (F \ {0}, *) is an abelian group with identity 1
13	- Distributive law holds
14	- `z3_solve.py prove "field_axioms"`
15
16	2. Field Extensions
17	- E is extension of F if F is subfield of E
18	- Degree [E:F] = dimension of E as F-vector space
19	- `sympy_compute.py minpoly "alpha" --var x` for minimal polynomial
20
21	3. Characteristic
22	- char(F) = smallest n > 0 where n*1 = 0, or 0 if none exists
23	- char(F) is 0 or prime
24	- For finite field: \|F\| = p^n where p = char(F)
25
26	4. Algebraic Elements
27	- alpha is algebraic over F if it satisfies polynomial with coefficients in F
28	- `sympy_compute.py solve "p(alpha) = 0"` for algebraic relations
29
30
31	## Tool Commands
32
33	### Z3_Field_Axioms
34	```bash
35	uv run python -m runtime.harness scripts/z3_solve.py prove "field_axioms"
36	```
37
38	### Sympy_Minpoly
39	```bash
40	uv run python -m runtime.harness scripts/sympy_compute.py minpoly "sqrt(2)" --var x
41	```
42
43	### Sympy_Solve
44	```bash
45	uv run python -m runtime.harness scripts/sympy_compute.py solve "x**2 - 2" --var x
46	```
47
48	## Key Techniques
49
50	From indexed textbooks:
51
52	- [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.
53	- [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.
54	- [Catergories for the working mathematician] Geometric Functional Analysis and Its Applications. Lectures in Abstract Algebra II. Lectures in Abstract Algebra III.
55	- [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)*.
56	- [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.
57
58	## Cognitive Tools Reference
59
60	See `.claude/skills/math-mode/SKILL.md` for full tool documentation.

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6 picks

Type