Solvability By Radicals Calculator

Example Data Table

Formula Used

This calculator applies a decision rule from Galois theory instead of one numeric formula. It combines degree rules, group structure, and optional invariants.

Solvable by radicals  ⟺  Gal(f/K) is a solvable group

Derived series:
G^(0) = G
G^(i+1) = [G^(i), G^(i)]
If G^(m) = {e} for some m, then G is solvable.

• Degree rule: all polynomials of degree 1–4 are solvable by radicals in characteristic 0.
• Abel–Ruffini: no general radical formula exists for degree 5 and above.
• Group rule: cyclic, abelian, dihedral, nilpotent, and many small groups are solvable.
• Non-solvable groups: A_n and S_n are non-solvable for n ≥ 5.
• Discriminant clue: if the discriminant is a square over Q, the Galois group lies inside A_n.

How to Use This Calculator

1. Enter the polynomial degree. This is the main branching input.
2. Select the family if your polynomial is binomial or cyclotomic.
3. Choose a known or suspected Galois group when available.
4. Add optional clues like irreducibility, discriminant square status, or derived length.
5. Click Submit to show the result above the form.
6. Use Download CSV for records or Download PDF for printable study notes.

Screening Value in Algebra Workflows

This calculator converts a difficult algebra topic into a structured and repeatable decision workflow. Users enter polynomial degree, family, suspected Galois group, and supporting clues, then receive a verdict with reasons and steps. That approach is valuable in classes, tutoring, and self study because it mirrors how mathematicians separate evidence collection from final proof. It also reduces avoidable mistakes, especially when a problem statement gives partial information only, not a full computation.

Degree Thresholds and Theorem Logic

The main theorem rule is built into the first branch. In characteristic zero, every polynomial of degree one through four is solvable by radicals, so the tool returns a direct solvable result. For degree five and above, the calculator stops using degree alone and shifts to group structure. This reflects Abel Ruffini correctly and prevents a common error: assuming every quintic is automatically non solvable by radicals in this setting often.

Using Group Evidence and Invariants

Group inputs drive the strongest conclusions. Solvable groups such as cyclic, abelian, dihedral, nilpotent, S3, A4, and S4 return solvable verdicts. Non solvable groups such as A5, PSL(2,5), and large alternating or symmetric groups return non solvable verdicts. Optional evidence fields improve interpretation: irreducibility helps identify the full group, discriminant square can suggest alternating containment, and derived length supports a finite commutator chain used for solvability arguments during initial screening clearly.

Reading the Output for Decisions

The output panel is designed for reasoning, not just labels. It reports a verdict, confidence level, key reasons, decision steps, warnings, and a compact input snapshot. This structure helps users audit their assumptions before sharing results or writing proofs. The complexity score is an educational index, not a theorem, combining degree with optional group order and derived length. It gives a quick way to compare cases in homework sets or notes after each submission.

Classroom, Audit, and Export Use Cases

The calculator also supports practical documentation. CSV export captures the verdict, assumptions, and reasoning in a portable format for spreadsheets or logs. PDF export is useful for handouts, revision packets, and review meetings. The example data table demonstrates common patterns: low degree equations are solvable, cyclotomic examples are abelian, and higher degree cases depend on group structure. This makes the page useful for both learning and consistent result reporting across teams.

FAQs