For an integer
n
≥
1
{\displaystyle n\geq 1}
, the minimal polynomial
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
of
2
cos
(
2
π
n
)
{\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)}
is the non-zero monic polynomial of degree
1
{\displaystyle 1}
for
n
=
1
,
2
{\displaystyle n=1,2}
and degree
1
2
φ
(
n
)
{\displaystyle {\frac {1}{2}}\varphi (n)}
for
n
≥
3
{\displaystyle n\geq 3}
with integer coefficients , such that
Ψ
n
(
2
cos
(
2
π
n
)
)
=
0
{\displaystyle \Psi _{n}\!\left(2\cos \left({\frac {2\pi }{n}}\right)\right)=0}
. Here
φ
(
n
)
{\displaystyle \varphi (n)}
denotes the Euler's totient function . In particular, for
n
≤
2
,
{\displaystyle n\leq 2,}
one has
Ψ
1
(
x
)
=
x
−
2
{\displaystyle \Psi _{1}(x)=x-2}
and
Ψ
2
(
x
)
=
x
+
2.
{\displaystyle \Psi _{2}(x)=x+2.}
For every n , the polynomial
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
is monic, has integer coefficients, and is irreducible over the integers and the rational numbers . All its roots are real ; they are the real numbers
2
cos
(
2
k
π
n
)
{\displaystyle 2\cos \left({\frac {2k\pi }{n}}\right)}
with k coprime with n and 1 ≤ k ≤ n (coprimality implies that k = n can occur only for n = 1 ). These roots are twice the real parts of the primitive n th roots of unity .
The polynomials
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
are typical examples of irreducible polynomials whose roots are all real and which have a cyclic Galois group .
Examples [ ]
The first few polynomials
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
are
Ψ
1
(
x
)
=
x
−
2
Ψ
2
(
x
)
=
x
+
2
Ψ
3
(
x
)
=
x
+
1
Ψ
4
(
x
)
=
x
Ψ
5
(
x
)
=
x
2
+
x
−
1
Ψ
6
(
x
)
=
x
−
1
Ψ
7
(
x
)
=
x
3
+
x
2
−
2
x
−
1
Ψ
8
(
x
)
=
x
2
−
2
Ψ
9
(
x
)
=
x
3
−
3
x
+
1
Ψ
10
(
x
)
=
x
2
−
x
−
1
Ψ
11
(
x
)
=
x
5
+
x
4
−
4
x
3
−
3
x
2
+
3
x
+
1
Ψ
12
(
x
)
=
x
2
−
3
{\displaystyle {\begin{aligned}\Psi _{1}(x)&=x-2\\\Psi _{2}(x)&=x+2\\\Psi _{3}(x)&=x+1\\\Psi _{4}(x)&=x\\\Psi _{5}(x)&=x^{2}+x-1\\\Psi _{6}(x)&=x-1\\\Psi _{7}(x)&=x^{3}+x^{2}-2x-1\\\Psi _{8}(x)&=x^{2}-2\\\Psi _{9}(x)&=x^{3}-3x+1\\\Psi _{10}(x)&=x^{2}-x-1\\\Psi _{11}(x)&=x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1\\\Psi _{12}(x)&=x^{2}-3\end{aligned}}}
Roots [ ]
The roots of
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
are given by
2
cos
(
2
π
k
n
)
{\displaystyle 2\cos \left({\frac {2\pi k}{n}}\right)}
,[1] where
1
≤
k
<
n
2
{\displaystyle 1\leq k<{\frac {n}{2}}}
and
gcd
(
k
,
n
)
=
1
{\displaystyle \gcd(k,n)=1}
. Since
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
is monic, we have
Ψ
n
(
x
)
=
∏
1
≤
k
<
n
2
gcd
(
k
,
n
)
=
1
(
x
−
2
cos
(
2
π
k
n
)
)
.
{\displaystyle \Psi _{n}(x)=\displaystyle \prod _{\begin{array}{c}1\leq k<{\frac {n}{2}}\\\gcd(k,n)=1\end{array}}\left(x-2\cos \left({\frac {2\pi k}{n}}\right)\right).}
Combining this result with the fact that the function
cos
(
x
)
{\displaystyle \cos(x)}
is even , we find that
2
cos
(
2
π
k
n
)
{\displaystyle 2\cos \left({\frac {2\pi k}{n}}\right)}
is an algebraic integer for any positive integer
n
{\displaystyle n}
and any integer
k
{\displaystyle k}
.
Relation to the cyclotomic polynomials [ ]
For a positive integer
n
{\displaystyle n}
, let
ζ
n
=
exp
(
2
π
i
n
)
=
cos
(
2
π
n
)
+
sin
(
2
π
n
)
i
{\displaystyle \zeta _{n}=\exp \left({\frac {2\pi i}{n}}\right)=\cos \left({\frac {2\pi }{n}}\right)+\sin \left({\frac {2\pi }{n}}\right)i}
, a primitive
n
{\displaystyle n}
-th root of unity. Then the minimal polynomial of
ζ
n
{\displaystyle \zeta _{n}}
is given by the
n
{\displaystyle n}
-th cyclotomic polynomial
Φ
n
(
x
)
{\displaystyle \Phi _{n}(x)}
. Since
ζ
n
−
1
=
cos
(
2
π
n
)
−
sin
(
2
π
n
)
i
{\displaystyle \zeta _{n}^{-1}=\cos \left({\frac {2\pi }{n}}\right)-\sin \left({\frac {2\pi }{n}}\right)i}
, the relation between
2
cos
(
2
π
n
)
{\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)}
and
ζ
n
{\displaystyle \zeta _{n}}
is given by
2
cos
(
2
π
n
)
=
ζ
n
+
ζ
n
−
1
{\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)=\zeta _{n}+\zeta _{n}^{-1}}
. This relation can be exhibited in the following identity proved by Lehmer, which holds for any non-zero complex number
z
{\displaystyle z}
:[2]
Ψ
n
(
z
+
z
−
1
)
=
z
−
φ
(
n
)
2
Φ
n
(
z
)
{\displaystyle \Psi _{n}\left(z+z^{-1}\right)=z^{-{\frac {\varphi (n)}{2}}}\Phi _{n}(z)}
Relation to Chebyshev polynomials [ ]
In 1993, Watkins and Zeitlin established the following relation between
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
and Chebyshev polynomials of the first kind.[1]
If
n
=
2
s
+
1
{\displaystyle n=2s+1}
is odd , then[verification needed ]
∏
d
∣
n
Ψ
d
(
2
x
)
=
2
(
T
s
+
1
(
x
)
−
T
s
(
x
)
)
,
{\displaystyle \prod _{d\mid n}\Psi _{d}(2x)=2(T_{s+1}(x)-T_{s}(x)),}
and if
n
=
2
s
{\displaystyle n=2s}
is even , then
∏
d
∣
n
Ψ
d
(
2
x
)
=
2
(
T
s
+
1
(
x
)
−
T
s
−
1
(
x
)
)
.
{\displaystyle \prod _{d\mid n}\Psi _{d}(2x)=2(T_{s+1}(x)-T_{s-1}(x)).}
Absolute value of the constant coefficient [ ]
The absolute value of the constant coefficient of
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
can be determined as follows:[3]
|
Ψ
n
(
0
)
|
=
{
0
if
n
=
4
,
2
if
n
=
2
k
,
k
≥
0
,
k
≠
2
,
p
if
n
=
4
p
k
,
k
≥
1
,
p
>
2
prime,
1
otherwise.
{\displaystyle |\Psi _{n}(0)|={\begin{cases}0&{\text{if}}\ n=4,\\2&{\text{if}}\ n=2^{k},k\geq 0,k\neq 2,\\p&{\text{if}}\ n=4p^{k},k\geq 1,p>2\ {\text{prime,}}\\1&{\text{otherwise.}}\end{cases}}}
Generated algebraic number field [ ]
The algebraic number field
K
n
=
Q
(
ζ
n
+
ζ
n
−
1
)
{\displaystyle K_{n}=\mathbb {Q} \left(\zeta _{n}+\zeta _{n}^{-1}\right)}
is the maximal real subfield of a cyclotomic field
Q
(
ζ
n
)
{\displaystyle \mathbb {Q} (\zeta _{n})}
. If
O
K
n
{\displaystyle {\mathcal {O}}_{K_{n}}}
denotes the ring of integers of
K
n
{\displaystyle K_{n}}
, then
O
K
n
=
Z
[
ζ
n
+
ζ
n
−
1
]
{\displaystyle {\mathcal {O}}_{K_{n}}=\mathbb {Z} \left[\zeta _{n}+\zeta _{n}^{-1}\right]}
. In other words, the set
{
1
,
ζ
n
+
ζ
n
−
1
,
…
,
(
ζ
n
+
ζ
n
−
1
)
φ
(
n
)
2
−
1
}
{\displaystyle \left\{1,\zeta _{n}+\zeta _{n}^{-1},\ldots ,\left(\zeta _{n}+\zeta _{n}^{-1}\right)^{{\frac {\varphi (n)}{2}}-1}\right\}}
is an integral basis of
O
K
n
{\displaystyle {\mathcal {O}}_{K_{n}}}
. In view of this, the discriminant of the algebraic number field
K
n
{\displaystyle K_{n}}
is equal to the discriminant of the polynomial
Ψ
n
(
x
)
{\displaystyle \Psi _{n}(x)}
, that is[4]
D
K
n
=
{
2
(
m
−
1
)
2
m
−
2
−
1
if
n
=
2
m
,
m
>
2
,
p
(
m
p
m
−
(
m
+
1
)
p
m
−
1
−
1
)
/
2
if
n
=
p
m
or
2
p
m
,
p
>
2
prime
,
(
∏
i
=
1
ω
(
n
)
p
i
e
i
−
1
/
(
p
i
−
1
)
)
φ
(
n
)
2
if
ω
(
n
)
>
1
,
k
≠
2
p
m
.
{\displaystyle D_{K_{n}}={\begin{cases}2^{(m-1)2^{m-2}-1}&{\text{if}}\ n=2^{m},m>2,\\p^{(mp^{m}-(m+1)p^{m-1}-1)/2}&{\text{if}}\ n=p^{m}\ {\text{or}}\ 2p^{m},p>2\ {\text{prime}},\\\left(\prod _{i=1}^{\omega (n)}p_{i}^{e_{i}-1/(p_{i}-1)}\right)^{\frac {\varphi (n)}{2}}&{\text{if}}\ \omega (n)>1,k\neq 2p^{m}.\end{cases}}}
References [ ]