Calculator
Formula used
If m | n, then GF(p^m) is a subfield of GF(p^n). Automorphisms fixing GF(p^m) form a cyclic Galois group of order n/m, generated by x ↦ x^(p^m).
How to use this calculator
- Enter a prime p and an extension degree n.
- Optionally set m to count automorphisms fixing GF(p^m).
- Pick a listing mode to display Frobenius-power mappings.
- Press Submit to show results above the form.
- Use CSV or PDF downloads to save your output.
Example data table
| p | n | m | m | n | |Aut(GF(p^n))| | |Aut(GF(p^n)/GF(p^m))| | Notes |
|---|---|---|---|---|---|---|
| 2 | 4 | 1 | Yes | 4 | 4 | All automorphisms over GF(2) are Frobenius powers. |
| 3 | 6 | 2 | Yes | 6 | 3 | Fixing GF(3²), the relative group has order 6/2. |
| 5 | 4 | 3 | No | 4 | 0 | No GF(5³) subfield inside GF(5⁴). |
Tip: When m does not divide n, the fixed-subfield count is not defined for finite fields.
Why finite-field automorphisms matter in computation
Finite fields GF(pn) appear in coding theory, cryptography, and algebra software. Knowing the automorphism count helps you predict symmetry, orbit sizes, and equivalence classes during searches. In GF(pn), every automorphism is a Frobenius power x ↦ xp, so the group is cyclic and has exactly n elements.
Interpreting the total automorphism count
The calculator reports |Aut(GF(pn))| = n. This is independent of the chosen prime p because the structure is controlled by the extension degree. Practically, if n = 8, there are 8 distinct field symmetries, and repeated Frobenius application cycles back to the identity after 8 steps.
Subfields and the “m divides n” data check
A key fact is that GF(pm) embeds as a subfield of GF(pn) exactly when m divides n. The form’s divisibility flag is a fast validity test: when it is “No,” the notion of automorphisms fixing GF(pm) does not apply. When it is “Yes,” the fixed-field subgroup has size n/m and is generated by x ↦ xpm.
Mapping list as reusable transformation data
Each listed mapping corresponds to a Frobenius-power exponent. In total mode the table shows k = 0…n−1, giving x ↦ xpk. In relative mode it shows k = 0…(n/m)−1 with exponent steps of m, giving x ↦ xpkm. These rows can be copied directly into notes or test scripts.
Exports for auditing, collaboration, and reproducibility
CSV export preserves numeric inputs, subgroup size, and the mapping list for spreadsheets or reports. The PDF export provides a compact summary plus the first few mappings for quick sharing. Together, these exports support reproducible computations when you compare field setups across experiments or teams.
FAQs
1) Why is the total automorphism count always n?
For GF(pn), every automorphism is a power of the Frobenius map x ↦ xp. Those powers form a cyclic group that returns to identity after exactly n steps, so the group has n elements.
2) What does the “m divides n” indicator mean?
It checks the subfield condition. GF(pm) is a subfield of GF(pn) if and only if m | n. If the indicator is “No,” the relative (fixed-subfield) automorphism count is not applicable.
3) Why can the relative count be n/m?
When m | n, automorphisms that fix GF(pm) are generated by x ↦ xpm. The subgroup consists of Frobenius powers spaced by m, giving exactly n/m distinct automorphisms.
4) Does the prime p affect the automorphism counts?
The counts depend on n (and on m for the relative subgroup), not on p. The prime only sets the characteristic and the concrete arithmetic inside the field, while the automorphism-group size stays the same.
5) What is shown in the mapping list table?
The table lists Frobenius-power maps. In total mode it shows x ↦ xpk for k = 0…n−1. In relative mode it shows x ↦ xpkm for k = 0…(n/m)−1.
6) Why might the mapping list be empty?
If relative mode is selected and m does not divide n, there is no valid fixed-subfield subgroup to list. The calculator still reports the total automorphisms, but suppresses the relative mapping table.