Study algebraic extensions using derivative tests degree splits. Track square-free structure and closure behavior clearly. Build reliable intuition with graphs, exports, and guided outputs.
Enter a characteristic, a degree split, and polynomial coefficients from highest power to constant term.
| Characteristic | Polynomial | [L:K] | e | Interpretation snapshot |
|---|---|---|---|---|
| 0 | x4 - 2 | 4 | 0 | Derivative is nonzero and the model is fully separable. |
| 2 | x4 + x2 + 1 | 4 | 1 | Derivative vanishes, indicating a p-power pattern and inseparable structure. |
| 3 | x6 + x3 + 2 | 9 | 1 | After one p-th root reduction, the square-free core becomes visible. |
Formal derivative: f'(x) = Σ i·aixi-1.
Repeated-root test: g(x) = gcd(f(x), f'(x)). A nonconstant gcd signals repeated algebraic roots.
Distinct-root degree metric: deg(f) − deg(g) when f' is not identically zero.
Degree split: [L:K] = [L:K]sep[L:K]ins.
Inseparable degree: [L:K]ins = pe in positive characteristic.
Separable degree: [L:K]sep = [L:K] / pe.
When f' = 0 in characteristic p, the tool repeatedly extracts p-th roots to expose the reduced square-free core.
1. Enter the field characteristic. Use 0 for characteristic zero or a prime number for positive characteristic fields.
2. Supply the modeled extension degree [L:K] and the inseparable exponent e.
3. Type coefficients from highest degree to constant term, separated by commas.
4. Choose the field type for interpretation notes.
5. Press the calculate button. The result block appears below the header and above the form.
6. Review derivative behavior, gcd degree, p-th root extractions, and separable versus inseparable degrees.
7. Export the result table as CSV or PDF for documentation.
It means the maximal separable algebraic extension inside a chosen algebraic closure. This tool does not build that closure explicitly. It estimates how much of a modeled extension behaves separably and how much is inseparable.
The formal derivative detects repeated algebraic roots. If gcd(f, f') is nonconstant, then the polynomial is not square-free over an algebraic closure, which is a classic warning sign in separability analysis.
That only happens in positive characteristic. It means every nonzero exponent is divisible by p, so the polynomial follows a p-power pattern. The tool then tries repeated p-th root reduction to expose a lower-degree core.
It is the square-free degree indicator. When f' is nonzero, it equals deg(f) − deg(gcd(f, f')). When f' is zero, the tool first reduces the polynomial by p-th root extraction and then measures the core.
No. A true explicit separable closure is typically infinite and highly nontrivial to construct. This page is a diagnostic degree-splitting tool designed for learning, checking, and documenting extension behavior.
If the entered extension degree is not divisible by pe, the inputs are inconsistent with a clean inseparable-degree factorization. The calculator still reports the numerical quotient so you can spot the mismatch immediately.
Yes. Finite fields are perfect, so every irreducible algebraic extension over them is separable. Repeated roots can still appear in reducible polynomials, but inseparable irreducible extensions do not occur there.
Yes. The derivative and gcd test works for reducible inputs too. Just remember that the output then describes the polynomial’s root structure and your entered degree split, not necessarily one irreducible field generator.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.