Cyclotomic polynomial Φ_n(x): minimal polynomial of primitive nth roots of unity. x^n−1 = Π_{d|n} Φ_d(x). Degree = φ(n). Φ_1=x−1, Φ_2=x+1, Φ_3=x²+x+1, Φ_4=x²+1, Φ_5=x⁴+x³+x²+x+1, Φ_6=x²−x+1.
How to Use
Enter n:
- Φ_n(x): Coefficients
- Degree: φ(n)
- Factorization: x^n−1 = Π Φ_d
Coefficient Mystery
For prime p: Φ_p = x^{p-1}+...+x+1 (all 1s). But Φ_n can have large coefficients! Φ_{105} is the first with a coefficient of −2. The maximum coefficient grows without bound but unpredictably.
Algebraic Significance
Φ_n is irreducible over ℚ (non-trivial!). The splitting field of Φ_n is ℚ(ζ_n) with Galois group (ℤ/nℤ)*. This connects cyclotomic polynomials to class field theory and algebraic number theory.
Step-by-Step Instructions
- 1Enter n.
- 2Compute Φ_n(x).
- 3Check degree = φ(n).
- 4See factorization.
- 5View roots.