Calculator
Formula used
- [K(α):K] = deg(m_α(x)) for algebraic α, where m_α is the minimal polynomial.
- [L:K] = [L:F] · [F:K] for a finite tower K ⊆ F ⊆ L.
- [GF(p^n):GF(p)] = n because GF(p^n) is an n-dimensional vector space over GF(p).
- [L:K] = dim_K(L) when L is finite-dimensional as a K-vector space.
How to use
- Select a method that matches your data.
- Enter degrees (or parameters) as positive integers.
- Press Submit to compute [L:K] and view steps.
- Use Download CSV or Download PDF to save results.
- Try the examples below to cross-check your inputs.
Example data table
| Scenario | Inputs | Degree | Reason |
|---|---|---|---|
| Simple extension | deg(mα) = 4 | 4 | Degree equals minimal polynomial degree. |
| Tower | [F:K]=2, [L:F]=3 | 6 | Multiply adjacent degrees: 2×3. |
| Finite field | p=5, n=3 | 3 | GF(5³) has dimension 3 over GF(5). |
| Basis size | Basis has 7 elements | 7 | Degree equals vector-space dimension. |
Minimal polynomial method in practice
When an algebraic element α is adjoined to a base field K, the degree of the extension is controlled by the minimal polynomial mα(x) over K. This calculator accepts that polynomial degree directly, so you can focus on modelling rather than lengthy reductions. In coursework, typical inputs come from proving irreducibility by Eisenstein, reduction modulo primes, or rational root tests, and then reading the resulting degree as [K(α):K].
Tower multiplication for chained constructions
Many extensions are built stepwise: K ⊆ F ⊆ L ⊆ M. If each intermediate degree is finite, the total degree is the product of adjacent degrees. The tower option lets you enter up to six factors, which is useful for splitting fields, radical adjunctions, and successive quadratic extensions. The worked steps clarify which factors were multiplied, helping you validate that each stage is finite and correctly oriented.
Finite fields and vector space dimension
For finite fields, the rule is clean: GF(p^n) is an n-dimensional vector space over GF(p). This tool records p and n, optionally checks that p is prime, and returns the extension degree n. That is ideal for coding theory, cryptography prerequisites, and counting arguments, where the vector space viewpoint also explains why every element satisfies x^{p^n} − x = 0. The same perspective supports counting subfields by divisors of n.
Basis size as a direct degree input
Sometimes the most reliable data is a basis. If you already have a K-basis for L, the extension degree equals the basis size. This occurs when you construct L as K[x]/(f), list residue classes, or compute a normal basis in finite fields. Entering the basis size bypasses polynomial computations while still producing a degree statement with an interpretation aligned to linear algebra. It is also helpful for checking independence during manual simplification.
Interpreting results and reporting outputs
After submission, the result appears immediately below the header so it can be copied into notes or proofs. CSV export stores the method, degree, and step list for auditing, while the PDF export creates a compact report. If your situation is transcendental or infinite-dimensional, the degree is not finite; in that case, use the calculator only for finite subextensions where the tower rule applies. Document the chosen method so later readers can reproduce your degree computation.
FAQs
1) What does the extension degree measure?
It measures the dimension of L as a vector space over K. A finite degree means every element of L can be expressed as a K-linear combination of finitely many basis elements.
2) When can I use the minimal polynomial option?
Use it for a simple algebraic extension K(α) when you know deg(mα(x)) over K. The degree equals that minimal polynomial degree for algebraic α.
3) How do I choose tower degrees correctly?
Enter degrees for each adjacent step in your chain, from the base field upward. The tool multiplies the provided factors, so reversing a step can produce the wrong total degree.
4) Why does GF(p^n) over GF(p) have degree n?
GF(p^n) is constructed as an n-dimensional vector space over GF(p). That dimension is exactly the field extension degree, independent of which irreducible polynomial was used.
5) Can this calculator handle infinite degrees?
No. Transcendental or infinite-dimensional extensions have infinite degree. Use the tool only for finite subextensions where a basis exists or the tower rule applies.
6) What do the CSV and PDF downloads include?
They include the selected method, the computed degree, a timestamp, and the worked steps shown on screen. This makes it easier to archive results alongside problem sets or reports.