Calculator Form
Use exact rational or integer coefficients for stronger symbolic conclusions.
Example Data Table
These examples show common splitting-field patterns over Q.
| Polynomial | Estimated Splitting Field | Extension Degree | Typical Group |
|---|---|---|---|
| x² − 2 | Q(√2) | 2 | C2 |
| x³ − 2 | Q(∛2, √Δ) | 6 | S3 |
| x³ − 3x + 1 | Q(α) | 3 | C3 / A3 |
| x⁴ − 5x² + 6 | Q(√2, √3) | 4 | V4 |
Formula Used
The calculator combines discriminants, rational-root tests, resolvent checks, and numerical roots. It is strongest over Q with exact coefficients.
Core formulas
Interpretation rules
A non-square discriminant gives Q(√Δ). A square discriminant means the polynomial already splits over Q.
For irreducible cubics, square discriminant usually gives degree 3. Otherwise the splitting field usually has degree 6.
Quartics use discriminant and resolvent behavior. General quartic outputs are sometimes candidate classifications, not formal proofs.
How to Use This Calculator
- Select degree 2, 3, or 4.
- Choose the base field: Q, R, or C.
- Enter coefficients from the highest power down to the constant term.
- Pick the precision and analysis mode that fit your problem.
- Press the calculate button to see the result summary above the form.
- Review the discriminant, candidate field, extension degree, root approximations, and notes.
- Use the CSV button for structured data export.
- Use the PDF button for a clean downloadable report.
FAQs
1) What is a splitting field?
A splitting field is the smallest field extension containing every root of a polynomial. Once inside that field, the polynomial factors completely into linear terms.
2) Are the results exact for every input?
Quadratic and many cubic cases are very reliable over Q with integer coefficients. General quartics can require symbolic factorization beyond this page, so some outputs are labeled as candidates.
3) Why does the discriminant matter here?
The discriminant detects repeated roots and often signals symmetry. For cubics and quartics over Q, whether the discriminant is a square can change the Galois-group prediction.
4) Which polynomials does this page support?
This calculator supports degree 2, 3, and 4 polynomials. Those are the most practical levels for fast symbolic classification with useful educational detail.
5) What happens if my polynomial has repeated roots?
A zero discriminant usually indicates repeated roots. In that case the splitting field may be smaller, because fewer distinct radicals or generators are needed.
6) Can I use decimal coefficients?
Yes, but exact rational tests become weaker. Integer or rational inputs give the strongest field descriptions, while decimal inputs can push the quartic logic toward heuristics.
7) Why does the quartic answer sometimes show multiple possibilities?
Quartic Galois classification can require finer factor tests. The page uses discriminants, rational roots, and the resolvent cubic, which sometimes narrows the answer to a candidate family.
8) Do the approximate roots affect the field result?
They mainly support interpretation and checking. The field estimate comes from algebraic tests first, while numerical roots help visualize real and complex behavior.