cyclotomic polynomial
For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of…For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial.
Meaning
1 sensesFor a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial.
For a prime number p, the pᵗʰ cyclotomic polynomial is #123;xᵖ-1#92;overx-1#125;#61;x#123;p-1#125;#43;x#123;p-2#125;#43;...#43;x²#43;x#43;1.
Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x#43;1.
Forms
plural: cyclotomic polynomials| Singular | cyclotomic polynomial |
|---|---|
| Plural | cyclotomic polynomials |
regular