Enter quintic coefficients
Use integer coefficients for the strongest theorem-based output. The page layout stays single-column, while the input controls respond as 3, 2, or 1 columns by screen size.
Example data table
| Example polynomial | Typical signal | Expected reading |
|---|---|---|
| x^5 - 1 | Rational root at x = 1 | Reducible over Q, therefore solvable by radicals. |
| x^5 - x - 1 | Irreducible with non-solvable cycle evidence | Usually reported as not solvable by radicals. |
| x^5 - 2 | Irreducible with compatible modular patterns | Often appears compatible with a solvable radical form. |
| (x + 1)^5 | Rational root at x = -1 | Reducible and immediately solvable by radicals. |
Formula used
f(x) = a₅x⁵ + a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀, with a₅ ≠ 0
Any rational root must have the form r = ±p/q, where p divides a₀ and q divides a₅.
If a prime p divides every nonleading coefficient, does not divide a₅, and p² does not divide a₀, then f(x) is irreducible over Q.
The calculator also checks f(x + k) for small integer shifts k. If a shifted form satisfies Eisenstein, the original quintic is irreducible over Q.
For a safe prime p, the factor degrees of f(x) modulo p match cycle lengths of a Frobenius element in the Galois group. Examples: 5, 1+4, 2+3, 1+1+3, 1+2+2, and 1+1+1+2.
A quintic over Q is solvable by radicals when its Galois group is solvable. For irreducible quintics of degree five, compatible solvable transitive groups are subgroups of AGL(1,5), such as C₅, D₅, and F₂₀.
How to use this calculator
- Enter the six integer coefficients from a₅ through a₀.
- Optionally set a custom graph interval and a prime list.
- Choose a shift limit if you want stronger shifted Eisenstein checks.
- Press Run Quintic Test.
- Read the result summary first, then inspect the modular factor table.
- Use the graph to study crossings, turning points, and scale.
- Export the report with CSV or PDF when needed.
FAQs
1) What does solvable by radicals mean?
It means every root can be built from rational numbers using finitely many additions, subtractions, multiplications, divisions, and nested radicals.
2) Why can a reducible quintic still be solvable?
If the quintic factors into lower-degree pieces, each factor has degree at most four or can be handled separately. Lower-degree factors admit radical formulas.
3) Does no rational root mean the quintic is unsolvable?
No. Some solvable quintics have no rational roots. That is why this page also checks irreducibility and modular cycle evidence.
4) Why are prime moduli useful here?
Factoring modulo suitable primes reveals cycle types inside the Galois group. Those cycle types help rule out or support solvable group structures.
5) What is Eisenstein’s criterion doing?
It gives a fast irreducibility proof. One well-chosen prime can certify that the quintic cannot factor over the rational numbers.
6) Why can the result be inconclusive?
Exact Galois groups are delicate. The tested primes may miss decisive patterns, or the polynomial may factor over Q without showing a rational root.
7) Should I use decimal coefficients?
Integer coefficients are best. The theorem-based tests on this page are designed for whole-number arithmetic and modular reductions.
8) What does the graph add to the algebraic test?
The plot shows y = f(x) across the chosen interval. It helps you inspect crossings, turning behavior, and scale, but it does not prove solvability by itself.