Map subfields inside finite extensions with confidence. Check divisors and tower degrees instantly. Export results and reuse them in proofs.
| d | Field | [GF(2^d):GF(2)] | [GF(2^6):GF(2^d)] |
|---|---|---|---|
| 1 | GF(2^1) | 1 | 6 |
| 2 | GF(2^2) | 2 | 3 |
| 3 | GF(2^3) | 3 | 2 |
| 6 | GF(2^6) | 6 | 1 |
Finite fields appear in coding, cryptography, and computational algebra because every field with p^n elements is isomorphic to GF(p^n). A key structural fact is that subfields are not arbitrary: they are indexed by the divisors of n. This makes intermediate-field discovery a clean arithmetic task rather than a symbolic one. The calculator turns that theorem into a practical checklist for practitioners.
If d divides n, then GF(p^d) sits inside GF(p^n), and there is exactly one such subfield up to equality. Intermediate fields between GF(p^m) and GF(p^n) exist precisely when m divides d and d divides n. So the search reduces to filtering the divisor set of n by a divisibility constraint from m. The output is a degree-ordered chain you can compare with your notes.
The number of intermediate fields equals the count of integers d satisfying m|d|n. When m=n, the count is one, meaning no proper intermediate field. When m=1, you obtain all subfields of GF(p^n). This count helps you sanity-check results, design example exercises, and estimate how many entries will appear before exporting to CSV or PDF.
In Reed–Solomon style constructions, subfields define trace and norm maps used for folding symbols and building constraints. In cryptographic protocols, choosing parameters sometimes requires avoiding small subfields to prevent subfield attacks. In implementation work, knowing the subfield degrees guides how you select bases, build embeddings, and reason about element representations across extension layers.
This tool assumes finite fields only; it does not compute explicit polynomials, generators, or embedding maps. Still, the degree list is often what you need first: it tells you which intermediate extensions are possible and which are impossible. If you later require explicit constructions, you can pair the degrees with an irreducible polynomial choice and a concrete basis. Use the example table to validate your intuition quickly. For study, try n=12 and m=3, then verify the divisor filter manually on paper.
It means a finite field GF(p^d) that lies between a smaller subfield GF(p^m) and a larger field GF(p^n), with m|d|n.
In finite fields, subfields correspond exactly to degrees d that divide n. Each such degree determines a unique subfield inside GF(p^n).
If m=n, there is only the top field. Also, if n is prime and m=1, the only subfields are GF(p) and GF(p^n).
No. It reports which degrees are possible. Constructing explicit embeddings needs an irreducible polynomial choice and a basis.
It is the number of degrees d that satisfy m|d|n. It counts subfields, not distinct generators or representations.
Not reliably. Number-field intermediate extensions depend on polynomial factorization and Galois theory data, not just divisibility.
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.