Calculator inputs
Use the simple criterion for one generator, the splitting-field criterion for full root closure, or the Galois shortcut for finite separable extensions.
Example data table
| Extension | Polynomial | Degree | Roots in K | Total distinct roots | Normal? |
|---|---|---|---|---|---|
| Q(√2) / Q | x² − 2 | 2 | 2 | 2 | Yes |
| Q(∛2) / Q | x³ − 2 | 3 | 1 | 3 | No |
| Q(∛2, ω) / Q | x³ − 2 | 6 | 3 | 3 | Yes |
| Q(ζ₅) / Q | Φ₅(x) | 4 | 4 | 4 | Yes |
Formula used
1. Simple extension test
If K = F(α) is finite and separable, then K/F is normal exactly when the minimal polynomial mα,F(x) splits completely in K[x].
2. Root coverage
Coverage = (roots contained in K ÷ total distinct conjugate roots) × 100%
3. Splitting field rule
If K is the splitting field of one or more polynomials from F[x], then K/F is normal by definition.
4. Finite separable Galois shortcut
For a finite separable extension, |AutF(K)| = [K:F] if and only if K/F is Galois, hence normal.
How to use this calculator
- Enter a label for the extension so your exports stay clear.
- Choose the method matching the theorem you actually know.
- For a single generator, type the minimal polynomial degree and count how many conjugate roots lie inside the extension.
- For a splitting field, enter the total roots required and how many are already inside K.
- If you know degree and automorphism count, use the Galois shortcut for a fast finite separable check.
- Press Check normality to show the result above the form, review the summary table, and inspect the Plotly graph.
- Use the CSV and PDF buttons to save the current result and recent history.
Recent calculations
The page stores up to 20 recent checks in the current session.
| Time | Extension | Method | Status | Galois check |
|---|---|---|---|---|
| No calculations saved yet. | ||||
FAQs
1. What is a normal extension?
A finite extension K/F is normal when every irreducible polynomial in F[x] that has one root in K splits completely over K. Equivalently, K contains all conjugates of the relevant generators.
2. Why can Q(∛2)/Q fail this test?
The polynomial x³ − 2 has three roots in an algebraic closure, but Q(∛2) contains only the real cube root. Missing complex conjugates prevent full splitting, so the extension is not normal.
3. Does normal always mean Galois?
No. Galois means normal and separable together. Over perfect fields such as Q, finite algebraic extensions are separable, so the distinction often disappears in common examples.
4. What does the automorphism test measure?
For a finite separable extension, the number of F-automorphisms cannot exceed [K:F]. Equality proves the extension is Galois, which immediately gives normality.
5. Why does the calculator ask about separability?
Separability matters because the clean root-coverage and automorphism criteria are strongest for finite separable extensions. Inseparable cases may need extra analysis using p-power polynomials and repeated roots.
6. Can this page check splitting fields directly?
Yes. If K is truly the splitting field of base-field polynomials and your root counts are complete, the extension is normal by definition. The calculator lets you test that setup explicitly.
7. What is the Plotly graph showing?
The graph summarizes supporting evidence such as root coverage, separability support, and automorphism coverage. It is a visual aid for comparison, not a substitute for the theorem itself.
8. Which inputs matter most for a reliable result?
Accurate root counts matter first. A correct minimal polynomial or splitting-field description is crucial. Automorphism counts are powerful confirmations, but only when the extension is finite and separable.