noun
Meaning 1
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.
Definition source: English Wiktionary via Wiktextract
Topics: algebra, mathematics, sciences
Examples
- 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.
Meaning relationships
Synonyms: none provided
Antonyms: none provided