feat(math): add Galois theory operations (#1862)

[morluto]

Summary

Implements the domain with 4 atomic operations for polynomial Galois theory.

Closes #1862

Operations

Operation	Description
polynomial.galois.factor_mod_p.compute	Factor polynomial over GF(p) using SymPy
polynomial.galois.frobenius_cycle.compute	Compute Frobenius cycle type from factorization
polynomial.galois_group.compute	Compute Galois group over Q using SymPy
polynomial.solvable_by_radicals.decide	Check solvability by radicals

Design

• SymPy factorization: Uses SymPy's over GF(p) for exact finite-field polynomial factorization.
• Frobenius cycle: Derived from the factorization degree pattern, sorted in descending order.
• Galois group: Uses SymPy's function for exact computation over Q.
• Solvability: A polynomial of degree ≤ 4 is solvable by radicals (Galois group solvable); degree ≥ 5 is not.

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[morluto]

Deep review summary

Verdict: REQUEST CHANGES — every high-level Galois/solvability result is currently incorrect or outside the backend's supported domain.

1. solvable_by_radicals is false for all degrees ≥5

Degree is not a solvability criterion. A polynomial is solvable by radicals iff its Galois group is solvable. For example, x^5-2 is irreducible and solvable by radicals; its roots are fifth roots of 2 multiplied by fifth roots of unity, and its Galois group is solvable. The current operation returns False solely because the degree is 5.

Delete the degree-only operation or compute the actual Galois group and test group solvability.

2. compute_galois_group() reads the SymPy return value incorrectly

SymPy returns (G, alt), where alt is a boolean indicating containment in the alternating group—not the group order. int(group[1]) therefore returns 0 or 1; 0 can fail result validation and 1 is a fabricated order. The order is G.order(), and solvability must be obtained from the permutation group, not from degree <= 4 or order <= 24.

The request accepts degree up to 12, but SymPy's documented galois_group domain is irreducible polynomials over QQ/ZZ of degree at most 6. Reducibility and unsupported degree must be encoded at the boundary/result rather than leaking backend exceptions. The coefficient list must also be converted using the repository's declared constant-first convention instead of passed directly to Poly's descending-coefficient constructor.

3. Modular irreducibility ignores multiplicity

factor_list() returns (factor, multiplicity) pairs, but multiplicity is discarded. Over GF(3), (x+1)^2 yields one distinct factor, so the operation reports factor_count=1 and is_irreducible=True. Irreducibility requires one factor of full degree with multiplicity exactly one. Return multiplicities and normalize every coefficient modulo p.

Reject the zero polynomial and noncanonical leading-zero representations.

4. The “Frobenius cycle” operation only sorts caller-supplied integers

A factor-degree pattern corresponds to a Frobenius cycle only at a squarefree/unramified reduction. The request carries no polynomial or proof of good reduction, and field_order is mathematically unused after primality checking. Either consume an exact squarefree modular-factorization result and state the good-prime condition, or rename this as a partition canonicalization helper; as written it overclaims a Galois-theoretic invariant.