Galois Resolvent Builder Calculator

Calculator

Input mode

Choose how you want to enter data.

Resolvent definition

Both are standard, but use different shifts.

Rounding (digits)

Affects display and exports only.

A (x⁴ coefficient)

B (x³ coefficient)

C (x² coefficient)

D (x coefficient)

E (constant term)

p (y² coefficient)

q (y coefficient)

r (constant term)

Reset

Example data table

Each row shows an input quartic, its depressed parameters, and the Ferrari-form resolvent coefficients.

Example	Quartic	p	q	r	x³	x²	x	1
Example 1	1x^4 + 0x^3 + -10x^2 + 5x + -2	-10.0000	5.0000	-2.0000	1.0000	10.0000	8.0000	55.0000
Example 2	1x^4 + 2x^3 + -3x^2 + -1x + 1	-4.5000	3.0000	0.5625	1.0000	4.5000	-2.2500	-19.1250
Example 3	3x^4 + 0x^3 + -6x^2 + 3x + -1	-2.0000	1.0000	-0.3333	1.0000	2.0000	1.3333	1.6667

Note: the displayed coefficients correspond to u³ - p u² - 4 r u + (4 p r - q²) = 0.

Formula used

1) Normalize the quartic

For A x⁴ + B x³ + C x² + D x + E = 0 with A ≠ 0, divide by A:

x⁴ + a₃ x³ + a₂ x² + a₁ x + a₀ = 0, where a₃=B/A, a₂=C/A, a₁=D/A, a₀=E/A.

2) Depress the quartic (remove the cubic term)

Substitute x = y - a₃/4 to obtain:

y⁴ + p y² + q y + r = 0

with

p = a₂ − 3a₃²/8
q = a₁ + a₃³/8 − a₃ a₂/2
r = a₀ − 3a₃⁴/256 + a₃² a₂/16 − a₃ a₁/4

3) Build the resolvent cubic

Two popular versions are available in this calculator:

Ferrari form: u³ − p u² − 4 r u + (4 p r − q²) = 0
Descartes form: z³ + 2 p z² + (p² − 4 r) z − q² = 0

How to use this calculator

Select an input mode: general quartic coefficients, or a depressed quartic.
Choose the resolvent definition you want to build.
Enter your coefficients and click Build resolvent.
Review the parameters p, q, r, the resolvent polynomial, and numeric real roots.
Use Download CSV or Download PDF to export results.

Resolvent objective and inputs

A resolvent cubic compresses quartic structure into a third‑degree polynomial whose roots encode key pairing information among the quartic roots. This calculator starts from either general coefficients or an already depressed quartic, then standardizes the model so results can be compared across problems. Because coefficients often come from measurement, optimization, or exam exercises, the tool emphasizes transparent intermediate values and consistent rounding rather than hidden symbolic shortcuts for study.

Depressed quartic parameters

For a general quartic, the first data transformation is normalization to a monic polynomial, followed by the shift x = y − a₃/4 that removes the cubic term. The resulting parameters p, q, and r summarize curvature, asymmetry, and offset in the depressed form y⁴ + p y² + q y + r = 0. Storing these values helps because many derivations and checks depend only on p, q, and r.

Ferrari and Descartes constructions

Two constructions are implemented. The Ferrari form u³ − p u² − 4 r u + (4 p r − q²) = 0 appears when completing squares in Ferrari’s method. The Descartes form z³ + 2 p z² + (p² − 4 r) z − q² aligns with symmetric sums of root pairings. Both are valid resolvents; select one based on the derivation you plan to use.

Interpreting resolvent roots

The displayed real roots are computed numerically using a stable depressed‑cubic workflow. The discriminant sign indicates whether the cubic has one real root or three real roots, which can guide expectations for factorization patterns of the quartic over the reals. In symbolic contexts, treat these values as approximations that suggest candidate expressions. In applied contexts, they serve as consistency checks for algebraic steps and rounding sensitivity.

Reporting and reproducibility

Professional work benefits from reproducible records. The CSV export captures inputs, p‑q‑r parameters, the chosen resolvent definition, coefficients, and the computed real roots in a format suitable for spreadsheets or notebooks. The PDF export creates a compact report that can be attached to assignments, reviews, or internal notes. When comparing scenarios, keep rounding digits consistent, and document which resolvent form you selected so collaborators can replicate the same intermediate values.

FAQs

1) What does the resolvent cubic represent?

It is a derived cubic whose roots encode structured combinations of quartic roots. Resolvent roots help reduce quartic solving to lower-degree steps and support reasoning about symmetric pairings and factorization behavior.

2) Why do we “depress” the quartic first?

Removing the cubic term via a simple shift simplifies the algebra. The depressed parameters p, q, and r concentrate the essential shape information, making the resolvent formulas shorter and more comparable across different quartics.

3) Which resolvent definition should I choose?

Choose Ferrari form if you are following completion-of-squares derivations or Ferrari-style solution steps. Choose Descartes form if your notes focus on symmetric sums of root pairings or alternate reduction pathways.

4) Why are only real resolvent roots shown?

The calculator focuses on quick, interpretable numeric checks. Real roots are often enough to validate workflows or explore real-factor patterns. Complex roots can be added later with a full complex solver if required.

5) Do resolvent roots determine the quartic roots directly?

Not by themselves. A resolvent root is an ingredient used to build auxiliary quantities that lead to quartic roots. The exact follow-up formulas depend on the method and on choosing consistent square-root branches.

6) How should I interpret rounding in exports?

Rounding affects presentation, not the underlying computation shown on the page. Use higher digits for close comparisons or large coefficients, and keep the same digits across runs when you need reproducible tables and reports.

Notes

This tool builds classical resolvent cubics for quartic polynomials; it does not attempt a full symbolic Galois group computation.
Numeric roots are provided for quick checking and visualization, not exact algebraic certification.
If your coefficients are large, consider increasing rounding digits for clearer exports.