cyclotomic polynomial

English

cyclotomic polynomial (plural cyclotomic polynomials)

(algebra) For a positive integer n, a polynomial whose roots are the primitive n^th roots of unity, so that its degree is the Euler's totient function of n. In symbols, letting ${\\displaystyle \\zeta _{n}=e^{i2\\pi /n}}$ be the first primitive n^th root of unity, then ${\\displaystyle \\Phi _{n}(x)=\\prod _{\\stackrel {1\\leq m<n}{\\gcd(n,m)=1}}(x-\\zeta _{n}^{m})}$ is the n^th such polynomial.
For a prime number ${\\displaystyle p}$ , the ${\\displaystyle p}$ ^th cyclotomic polynomial is ${\\displaystyle {x^{p}-1 \\over x-1}=x^{p-1}+x^{p-2}+...+x^{2}+x+1}$ .
Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing ${\\displaystyle x}$ with ${\\displaystyle x+1}$ .

单词	cyclotomic polynomial
释义	cyclotomic polynomial English Noun cyclotomic polynomial (plural cyclotomic polynomials) (algebra) For a positive integer n, a polynomial whose roots are the primitive n^th roots of unity, so that its degree is the Euler's totient function of n. In symbols, letting ${\\displaystyle \\zeta _{n}=e^{i2\\pi /n}}$ be the first primitive n^th root of unity, then ${\\displaystyle \\Phi _{n}(x)=\\prod _{\\stackrel {1\\leq m<n}{\\gcd(n,m)=1}}(x-\\zeta _{n}^{m})}$ is the n^th such polynomial. For a prime number ${\\displaystyle p}$ , the ${\\displaystyle p}$ ^th cyclotomic polynomial* is ${\\displaystyle {x^{p}-1 \\over x-1}=x^{p-1}+x^{p-2}+...+x^{2}+x+1}$ .* Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing ${\\displaystyle x}$ with ${\\displaystyle x+1}$ .
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