Calculator Inputs
Formula Used
A polynomial f(x) is separable when it has no repeated roots in an algebraic closure. A practical test is:
- g(x) = gcd(f(x), f′(x))
- If g(x) = 1, then f is square‑free and separable.
- If deg(g) ≥ 1, then f has a repeated factor and is not separable.
In prime characteristic, if f′(x) = 0, the polynomial is inseparable.
How to Use This Calculator
- Enter integer coefficients for your polynomial.
- Select the coefficient order you are providing.
- Pick characteristic 0 or a prime characteristic p.
- Press Submit & Test to see results above.
- Download CSV or PDF for notes and submissions.
Example Data Table
| Example | Coefficients | Characteristic | Expected |
|---|---|---|---|
| x^3 - x + 1 | 1, 0, -1, 1 | 0 | Separable (typical over characteristic 0) |
| (x-1)^2(x+2) = x^3 +0x^2 -3x +2 | 1, 0, -3, 2 | 0 | Not separable (repeated root at x=1) |
| x^5 + 1 over p=5 | 1, 0, 0, 0, 0, 1 | 5 | Not separable (derivative is 0 mod 5) |
| x^4 + 2x + 3 over p=7 | 1, 0, 0, 2, 3 | 7 | Separable when gcd is 1 |
Professional Notes
Operational purpose and typical use cases
Separable polynomials matter whenever you count distinct roots, build splitting fields, or factor without multiplicities. This calculator targets classroom algebra, coding theory worksheets, and finite field exercises where repeated factors change root counts. In characteristic zero, separability aligns with being square‑free, so the gcd test quickly distinguishes clean factorisations from those containing repeated linear or irreducible factors.
Data flow from coefficients to a verdict
The input is a coefficient vector, interpreted as highest degree to constant, then trimmed to remove leading zeros. The program computes the formal derivative term by term, using power multipliers, and applies Euclidean division to obtain gcd(f, f′). Degree tracking is reported because a gcd of degree zero corresponds to the constant polynomial 1, the threshold for separability.
Characteristic selection and edge conditions
Prime characteristic changes arithmetic and can create the special case f′(x)=0. This occurs when every exponent is a multiple of p, such as x^5+1 in p=5. In that situation the polynomial is inseparable, and the tool flags it explicitly. When the derivative is nonzero, the same gcd criterion works in the chosen field, with coefficients reduced modulo p.
Interpreting gcd output for factor structure
If gcd(f, f′) has positive degree, it shares a nontrivial factor with f. That shared factor corresponds to repeated roots in an algebraic closure and repeated irreducible factors in the factorisation of f. Practically, the gcd isolates the “square part” of the polynomial. When the gcd is 1, each irreducible factor appears once, supporting distinct‑root assumptions used in proofs and computations.
Reporting, exports, and reproducible study records
The summary panel formats f(x), f′(x), the gcd, its degree, and a short verdict. CSV export stores the same fields in a two‑column key/value layout for spreadsheets, while the PDF export captures a single‑page snapshot suitable for assignments. For consistent records, keep the same coefficient order, label, and characteristic across runs, and attach exports alongside your worked factorisations. When comparing two polynomials, run both under the same p to avoid mixing fields. If your input contains leading zeros, trimming changes the displayed degree, so check the highest nonzero coefficient before interpreting results. in practice.
FAQs
1) What does “separable” mean here?
It means the polynomial has no repeated roots in an algebraic closure of the chosen field. Equivalently, the factorisation has no repeated irreducible factor, so each root contributes multiplicity one.
2) Why is gcd(f, f′) the right test?
A repeated root makes both f and its derivative vanish at that root, so they share a factor. When the gcd is 1, no such shared factor exists, so the polynomial is square‑free.
3) What if the derivative is the zero polynomial?
In prime characteristic, the derivative can become identically zero when every exponent is a multiple of p. That situation indicates inseparability, and the calculator reports it as not separable.
4) Do I need to factor the polynomial first?
No. The gcd computation avoids explicit factorisation and still detects repeated factors. Factoring can be used later to interpret which factor repeats, but it is not required for the separability verdict.
5) How should I enter coefficients correctly?
Provide integers separated by commas or spaces. Choose whether your list runs from highest degree to constant or the reverse. Leading zeros are allowed, but trimming may reduce the effective degree shown.
6) What do the CSV and PDF downloads contain?
Both exports include the characteristic, f(x), f′(x), gcd(f, f′), gcd degree, and the final verdict. CSV suits spreadsheets, while the PDF is a compact single‑page record for sharing.