Last Updated：2025/12/04
(algebra) 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=ei 2𝜋/n be the first primitive nᵗʰ root of unity, then 𝛷ₙ(x)=∏_( stackrel )1
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cyclotomic polynomial
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Source Word
cyclotomic polynomial
Noun
(algebra)
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.
Japanese Meaning
巡回多項式：正の整数 n に対して、原始的な n 次の単位根をすべて根とする多項式であり、その次数は n のオイラーのトーシェント関数（φ(n)）によって定まる。
Sense（1）
(algebra)
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.
（ plural ）
