1.NounFor a positive integer n, a polynomial whose roots are the primitive n roots of unity, so that its degree is Euler's totient function of n. That is, letting
ζ
n
=
e
i
2
π
/
n
{\displaystyle \zeta _{n}=e^{i2\pi /n}}
be the first primitive n root of unity, then
Φ
n
(
x
)
=
∏
gcd
(
n
,
m
)
=
1
1
≤
m
<
n
(
x
−
ζ
n
m
)
{\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq m<n}{\gcd(n,m)=1}}(x-\zeta _{n}^{m})}
is the n such polynomial.
"For a prime number
p
{\displaystyle p}
, the
p
{\displaystyle p}
cyclotomic polynomial is
x
p
−
1
x
−
1
=
x
p
−
1
+
x
p
−
2
+
.
.
.
+
x
2
+
x
+
1
{\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
x
{\displaystyle x}
with
x
+
1
{\displaystyle x+1}
."