Calculator Inputs
Best exact classification occurs for primitive integer polynomials of degree 2, 3, or 4 over Q.
Example Data Table
| Polynomial | Degree | Discriminant | Likely group | Estimated closure degree | Comment |
|---|---|---|---|---|---|
| x2 - 5 | 2 | 20 | C2 | 2 | Irreducible quadratic with non-square discriminant. |
| x3 - 2 | 3 | -108 | S3 | 6 | Irreducible cubic with non-square discriminant. |
| x4 - 2 | 4 | -2048 | D4 or C4 | 8 or 4 | Imprimitive quartic case after resolvent reduction. |
| x4 - 5x2 + 6 | 4 | 24 | V4 | 4 | Product of two irreducible quadratics over Q. |
Formula Used
1) Discriminant formulas
For a quadratic ax² + bx + c, the discriminant is Δ = b² - 4ac.
For a cubic ax³ + bx² + cx + d, the calculator uses Δ = b²c² - 4ac³ - 4b³d - 27a²d² + 18abcd.
For a quartic ax⁴ + bx³ + cx² + dx + e, the full quartic discriminant formula is used directly.
2) Separable test
Over Q, a polynomial is separable when Δ ≠ 0. A zero discriminant means repeated roots, so the closure analysis changes immediately.
3) Cubic closure rule
For an irreducible cubic over Q:
- If Δ is a square, the Galois group is A3 ≅ C3.
- If Δ is not a square, the Galois group is S3.
4) Quartic resolvent
For the monic quartic x⁴ + Bx³ + Cx² + Dx + E, the cubic resolvent is y³ - Cy² + (BD - 4E)y + (4CE - B²E - D²).
Its reducibility, together with the discriminant square test, separates the usual quartic candidates: S4, A4, V4, and the imprimitive case D4 or C4.
5) Closure degree principle
For irreducible cases, the estimated Galois closure degree equals the likely order of the computed symmetry group. For reducible inputs, the closure is the compositum of the nontrivial factor splitting fields.
How to Use This Calculator
- Select the polynomial degree from 2, 3, or 4.
- Enter the coefficients from the highest power down to the constant term.
- Use primitive integer inputs whenever possible for the strongest exact rules.
- Choose a display precision for formatted outputs.
- Press Find Galois Closure to compute the discriminant, separability, factor hints, and closure estimate.
- Read the result cards first, then inspect the detailed table and interpretation notes.
- Use the Plotly graph to view the numerical roots in the complex plane.
- Download the summary as CSV or PDF for class notes, proofs, or worksheets.
FAQs
1) What does this calculator actually find?
It estimates the splitting-field symmetry of a degree 2, 3, or 4 polynomial over Q. It reports discriminant data, separability, reducibility clues, and a likely Galois closure degree.
2) Is the quartic result always exact?
Not always. Exact quartic separation is strongest for monic integer inputs. Some nonmonic or noninteger quartics are still analyzed well, but their final group may remain a candidate set.
3) Why is the discriminant so important?
The discriminant detects repeated roots and helps decide whether key square tests hold. For cubics and quartics, it is one of the fastest routes to the likely Galois group.
4) What does a square discriminant mean?
In these degree rules, a square discriminant usually signals that the Galois group lies inside an alternating-type subgroup. For irreducible cubics, it gives A3. For quartics, it sharply narrows the choices.
5) Why does the chart use numerical roots?
The plot is meant for visualization. It places approximate roots in the complex plane so you can see conjugate structure, repeated behavior, and geometric splitting patterns quickly.
6) Can I use decimal coefficients?
Yes. The calculator accepts decimals, but exact factor and group rules are strongest when coefficients are primitive integers. Decimal input may move the result into a more heuristic mode.
7) What does closure degree represent?
It represents the degree of the Galois closure, equivalently the degree of the splitting field over Q in the intended cases. For irreducible inputs, it often matches the order of the likely group.
8) Can this replace a full computer algebra system?
No. It is a strong instructional calculator for low-degree polynomials, not a complete symbolic algebra engine. It gives practical closure insight quickly without requiring heavy external algebra packages.