Cyclotomic Polynomial Calculator

Analyze primitive roots, degrees, and coefficient patterns fast. Check divisor products and symbolic expansions effortlessly. Built for clear algebra work across screens and classrooms.

This calculator builds the cyclotomic polynomial Φn(x), reports its degree, lists its coefficients, evaluates it at a chosen value, and explains the divisor structure behind the result.

Calculator Input

Use positive integers up to 60 for stable symbolic output and easy reading.

Formula Used

The calculator applies the recursive identity Φn(x) = (xn - 1) / ∏ Φd(x) over all proper divisors d of n. This creates the exact polynomial with integer coefficients.

The degree follows deg(Φn) = φ(n), where φ is Euler’s totient function. The roots are the primitive n-th roots of unity, so each valid exponent k satisfies gcd(k, n) = 1.

How to Use This Calculator

  1. Enter the index n for the desired cyclotomic polynomial.
  2. Optionally add a numeric x value for direct evaluation.
  3. Choose extra output for primitive roots or a quick value table.
  4. Press Compute Polynomial to place the result above the form.
  5. Use the CSV or PDF buttons to save the generated output.

Example Data Table

n Φn(x) Degree Notes
3 x2 + x + 1 2 Primitive cube roots of unity.
5 x4 + x3 + x2 + x + 1 4 Prime index gives a simple geometric sum.
8 x4 + 1 4 Power-of-two case stays compact.
10 x4 - x3 + x2 - x + 1 4 Alternating coefficients appear here.

FAQs

1. What does a cyclotomic polynomial represent?

It is the minimal polynomial whose roots are the primitive n-th roots of unity. These polynomials are central in algebra, number theory, and field extensions.

2. Why is the degree equal to Euler’s totient function?

The degree counts primitive n-th roots of unity. That count is exactly φ(n), because each valid exponent k must be relatively prime to n.

3. Why are the coefficients integers?

Cyclotomic polynomials are built from factorization of x^n − 1 and simplify into monic polynomials with integer coefficients. This is a standard algebraic result.

4. Can this calculator handle prime and composite n values?

Yes. It works for both. Prime inputs often produce simple sums, while composite values can create alternating or sparse coefficient patterns.

5. What is the practical use of evaluating Φn(x)?

Evaluating the polynomial helps test identities, compare factor behavior, inspect sign changes, and study algebraic patterns at selected numeric points.

6. Why limit the input size?

Large n values can create high-degree outputs with many coefficients. The limit keeps calculations readable, reliable, and fast in normal browser sessions.

7. Are primitive root exponents the same as roots themselves?

Not exactly. The exponents identify which powers of ζn produce primitive roots. They are a compact way to describe the full root set.

8. Does the calculator use Möbius ideas?

Yes indirectly. Cyclotomic identities are closely tied to Möbius inversion. This version uses recursive exact division, which gives the same polynomial result.