Enter Cyclotomic Parameters
Example Data Table
| n | φ(n) | Φn(x) | |disc| | N(1-ζn) |
|---|---|---|---|---|
| 5 | 4 | x4 + x3 + x2 + x + 1 | 125 | 5 |
| 8 | 4 | x4 + 1 | 256 | 2 |
| 9 | 6 | x6 + x3 + 1 | 19683 | 3 |
| 12 | 4 | x4 - x2 + 1 | 144 | 1 |
Formula Used
Field degree: [ℚ(ζ_n):ℚ] = φ(n) = n∏(1 - 1/p) over distinct primes p | n.
Primitive power order: for ζ_n^k, the order is n / gcd(n,k). Its minimal polynomial is Φ_m(x) with m = n / gcd(n,k).
Discriminant: for n > 2, disc(ℚ(ζ_n)) = (-1)^{φ(n)/2} n^{φ(n)} / ∏_{p|n} p^{φ(n)/(p-1)}.
Trace of a power: Tr(ζ_n^k) equals the Ramanujan sum c_n(k), computed here by μ(m)φ(n)/φ(m) where m = n/gcd(n,k).
Norm of 1-ζ_n: it equals p when n = p^a, and equals 1 otherwise.
Cyclotomic polynomial expansion: the page expands manageable cases using exact polynomial division from x^n - 1 = ∏_{d|n} Φ_d(x).
How to Use This Calculator
- Enter a positive integer n to define the cyclotomic field generated by an n-th primitive root of unity.
- Enter an optional exponent k to analyze the element ζnk.
- Choose a one-letter polynomial variable if you want polynomial output in a preferred symbol.
- Set a preview limit for reduced residues and an exact digit threshold for discriminant display.
- Keep the polynomial checkbox enabled when you want explicit Φn(x) for smaller orders.
- Press Calculate Cyclotomic Field to show the result below the header and above the form.
- Use Download CSV for data export and Download PDF for a printable summary.
Frequently Asked Questions
1. What does this calculator mainly compute?
It computes core invariants of ℚ(ζn): degree, discriminant, polynomial data, primitive root counts, subfield degree, and power behavior for ζnk.
2. Why does the degree equal φ(n)?
The n-th cyclotomic polynomial Φn(x) is irreducible over ℚ and has exactly φ(n) primitive n-th roots, so its degree is φ(n).
3. What does the exponent k change?
It changes the order of ζnk. That determines whether the chosen element is primitive, what subfield it generates, and the degree of its minimal polynomial.
4. Why is the polynomial sometimes not expanded?
Explicit cyclotomic polynomials can become very large. The calculator expands only manageable orders to keep the page responsive and the output readable.
5. What happens for n = 1 or n = 2?
Both cases give the rational field ℚ. Their degree is 1, the discriminant is 1, and the page still reports useful power and polynomial information.
6. What is the maximal real subfield?
For n > 2, the maximal real subfield is generated by ζn + ζn-1. Its degree is φ(n)/2.
7. What is special about N(1-ζ_n)?
This norm sharply detects prime powers. It equals the prime p when n = pa, and equals 1 for all other n greater than 1.
8. Can I use the output for proofs or coursework?
Yes, it is useful for checking computations, examples, and conjectures. For formal proofs, still cite standard algebra texts for theorems behind the formulas.