Calculator input
Use descending-degree coefficients. Example: 1,0,1,1 means x3 + x + 1.
Example data table
These examples use descending-degree coefficient input.
| Polynomial input | Meaning | Field | Expected verdict |
|---|---|---|---|
| 1,0,1,1 | x3 + x + 1 | GF(2) | Irreducible |
| 1,0,1 | x2 + 1 | GF(5) | Reducible |
| 1,0,1 | x2 + 1 | GF(3) | Irreducible |
| 1,0,1,0,1 | x4 + x2 + 1 | GF(2) | Reducible |
Formula used
Field setup: coefficients are reduced modulo p, where p is prime.
Normalization: if the leading coefficient is nonzero, the calculator scales the polynomial to a monic form. Irreducibility does not change under nonzero scaling inside a field.
Rabin test: let f(x) have degree n over GF(p).
1. The calculator checks whether x^(p^n) - x ≡ 0 (mod f(x)).
2. For every prime divisor q of n, it checks whether gcd(f(x), x^(p^(n/q)) - x) = 1.
The polynomial is irreducible exactly when both conditions hold. A root scan is also shown because any field root immediately gives a linear factor.
How to use this calculator
- Enter coefficients from highest degree to constant term.
- Choose a prime modulus p for the field GF(p).
- Submit the form to display the verdict above the form.
- Read the reduced polynomial, monic form, roots, and gcd screens.
- Inspect the coefficient graph for a quick visual summary.
- Download the result summary as CSV or PDF if needed.
Frequently asked questions
1) What kind of irreducibility does this calculator test?
It tests irreducibility over the finite field GF(p), where p is prime. That means coefficients are reduced modulo p first. A polynomial can be irreducible over one field and reducible over another, so the chosen modulus matters.
2) How should I enter coefficients?
Enter integers in descending-degree order, separated by commas or spaces. For example, 1,0,1,1 means x^3 + x + 1. Negative coefficients are accepted and automatically reduced modulo the chosen prime field.
3) Why does the calculator convert to a monic polynomial?
Multiplying by a nonzero field constant does not change irreducibility. Converting to monic form simplifies the polynomial arithmetic, keeps gcd results cleaner, and makes the Rabin test easier to present in a consistent way.
4) What does a root in GF(p) tell me?
A root a in GF(p) means x - a divides the polynomial. That immediately makes any degree 2 or higher polynomial reducible over that field. The root scan is therefore a quick diagnostic before reading the deeper Rabin checkpoints.
5) Why are degree 2 and degree 3 cases special?
Over a field, any reducible quadratic or cubic must have a linear factor, so it must have a root in the field. Therefore, for degree 2 or 3, finding no root already proves irreducibility.
6) Does this calculator factor the polynomial completely?
No. It is designed to test irreducibility and show the most useful checkpoints. If the result is reducible, that does not automatically provide a full factorization. It only proves that a nontrivial factor exists over the chosen field.
7) Why is the modulus restricted to prime values?
The arithmetic relies on inverses for every nonzero coefficient during polynomial division and gcd computation. That works cleanly in a field. Using a prime modulus guarantees the coefficient system is GF(p), where those inverses always exist.
8) What do the CSV and PDF downloads contain?
They export a compact summary of the tested field, degree, verdict, reduced polynomial, monic form, root information, and Rabin checkpoint status. This is useful for saving worked examples, validating classroom exercises, or documenting research notes.