Separable Degree Calculator

Tip: Extension mode uses [L:K]ₛ = [L:K] / [L:K]ᵢ. Polynomial mode uses the square-free part f / gcd(f,f').

Example data table

Saved results (for downloads)

What separable degree measures

Separable degree counts the “distinct” algebraic part of a finite extension. When an extension L/K has total degree [L:K], the calculator reports the separable factor [L:K]ₛ after removing any purely inseparable contribution. In practice, [L:K]ₛ tells you how many K‑embeddings of L land in a separable closure. It is the value used in many theorems about trace, norm, and splitting behavior.

When inseparable degree appears

Inseparability only occurs in positive characteristic. The calculator lets you enter the inseparable degree directly, or derive it as p^e using characteristic p and exponent e. For example, if [L:K]=12 and [L:K]ᵢ=4, then [L:K]ₛ=3. If p=3 and e=2, then p^e=9; with [L:K]=9 the separable degree becomes 1, indicating a purely inseparable situation.

Why polynomial square-free part matters

Polynomial mode focuses on repeated factors, because repeated roots correspond to inseparability in the derivative test. Over characteristic 0, a polynomial is square‑free exactly when gcd(f, f′)=1. The calculator computes g(x)=gcd(f(x), f′(x)), forms the square‑free part f/g, and reports its degree as a separable‑degree proxy. Coefficients may be integers or fractions, and the gcd is computed using exact rational arithmetic to avoid rounding errors. For f(x)=x⁴−2x²+1, the square‑free part is x²−1, so the reported separable degree is 2.

Typical degrees in worked scenarios

Use extension mode when degrees are known from theory or from tower laws. For towers K⊂M⊂L, the separable degrees multiply: [L:K]ₛ=[L:M]ₛ·[M:K]ₛ, while inseparable degrees multiply similarly. This calculator helps sanity‑check values by enforcing divisibility and highlighting whether an entered [L:K]ᵢ is a pure power of p. Over finite fields, algebraic extensions are separable, so [L:K]ᵢ should be 1 and the calculator will reflect that. That is useful when your notes claim p^e but the arithmetic disagrees.

Exporting results for documentation

Each submission is stored in a session history table so you can compare cases side‑by‑side. CSV export is convenient for spreadsheets and lab notebooks, while PDF export produces a clean snapshot for reports and audits. Because the result block appears above the form after submit, you can immediately capture the computed [L:K]ₛ along with intermediate values. Clearing history removes only the current session data and does not affect the calculator code.

FAQs

1) What inputs do I need for Extension mode?

Enter the total degree [L:K] and either the inseparable degree [L:K]ᵢ directly or (p, e) so that [L:K]ᵢ = p^e. The calculator then computes [L:K]ₛ automatically.

2) Why must the inseparable degree divide the total degree?

For finite extensions, the standard factorization is [L:K] = [L:K]ₛ · [L:K]ᵢ. If divisibility fails, the inputs cannot describe a valid decomposition, so the calculator stops and reports an error.

3) Can I set characteristic 0 with exponent e > 0?

No. In characteristic 0, the inseparable degree is 1 in the usual theory, so the calculator requires e = 0 whenever p = 0. Use direct input if you are modeling something nonstandard.

4) How are fractions handled in Polynomial mode?

You may enter coefficients as integers or fractions like 3/2. The gcd and division steps use exact rational arithmetic, so the square-free part and the reported degree are not affected by floating-point rounding.

5) Does Polynomial mode support positive characteristic?

This implementation uses the derivative square-free test as written for characteristic 0. In characteristic p, derivatives can vanish for p-th powers, so separability checks require additional rules not included here.

6) What is included in the CSV and PDF downloads?

Downloads are generated from the saved session history table. They include the timestamp, selected mode, a compact input summary, the separable degree, and short notes. Clear history if you want a fresh export set.

Formula used

• Finite extension: The separable degree is the total degree divided by the inseparable degree.
  [L:K]ₛ = [L:K] / [L:K]ᵢ (requires exact divisibility).
• Inseparable degree as a power: In characteristic p>0, the inseparable degree often has the form p^e.
  [L:K]ᵢ = p^e
• Polynomial square-free idea (characteristic 0): Repeated factors are detected via the gcd with the derivative.
  g(x)=gcd(f(x), f'(x)),  squareFree=f/g and we report deg(squareFree).