Début de la boite de navigation du chapitre
fin de la boite de navigation du chapitre
En raison de limitations techniques, la typographie souhaitable du titre, «
Polynômes annulateurs de nombres sous la forme cosinus ou tangente : Polynômes annulateurs de nombres de la forme 2cos(2kπ╱n) Polynômes annulateurs de nombres sous la forme cosinus ou tangente/Polynômes annulateurs de nombres de la forme 2cos(2kπ╱n) », n'a pu être restituée correctement ci-dessus.
Ci-dessous se trouvent, dans l’ordre croissant des degrés sur
Q
{\displaystyle \mathbb {Q} }
Q
(
d
)
{\displaystyle \mathbb {Q} ({\sqrt {d}})}
Q
{\displaystyle \mathbb {Q} }
entiers algébriques de la forme
2
cos
(
2
k
π
/
n
)
{\displaystyle 2\cos(2k\pi /n)}
0
<
k
<
n
/
2
{\displaystyle 0<k<n/2}
k premier avec n .Lorsque n est pair, la fraction
2
k
/
n
{\displaystyle 2k/n}
Rappelons (cf. chap. 1) que le degré (sur
Q
{\displaystyle \mathbb {Q} }
cos
2
k
π
n
{\displaystyle \cos {\frac {2k\pi }{n}}}
k
{\displaystyle k}
n
{\displaystyle n}
n
≥
3
{\displaystyle n\geq 3}
φ
(
n
)
2
{\displaystyle {\frac {\varphi (n)}{2}}}
≠
7
,
9
{\displaystyle \neq 7,9}
φ
(
n
)
{\displaystyle \varphi (n)}
Q
{\displaystyle \mathbb {Q} }
φ
(
n
)
4
{\displaystyle {\frac {\varphi (n)}{4}}}
Q
(
d
)
{\displaystyle \mathbb {Q} ({\sqrt {d}})}
(
Z
/
n
Z
)
×
/
{
±
1
}
{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }/\{\pm 1\}}
n est divisible par 8). De plus, l'entier
d
{\displaystyle d}
4
d
{\displaystyle 4d}
d
≢
1
mod
4
{\displaystyle d\not \equiv 1{\bmod {4}}}
n
{\displaystyle n}
discriminant de
Q
(
d
)
{\displaystyle \mathbb {Q} ({\sqrt {d}})}
divise celui de l'extension cyclotomique ℚ(ζn .
Nous éviterons les redondances en remarquant que si
X
m
+
a
m
−
1
X
m
−
1
+
a
m
−
2
X
m
−
2
+
⋯
+
a
0
{\displaystyle X^{m}+a_{m-1}X^{m-1}+a_{m-2}X^{m-2}+\dots +a_{0}}
K de
2
cos
2
k
π
n
{\displaystyle 2\cos {\frac {2k\pi }{n}}}
alors
X
m
−
a
m
−
1
X
m
−
1
+
a
m
−
2
X
m
−
2
+
⋯
+
(
−
1
)
m
a
0
{\displaystyle X^{m}-a_{m-1}X^{m-1}+a_{m-2}X^{m-2}+\dots +(-1)^{m}a_{0}}
K de son opposé,
2
cos
(
n
−
2
k
)
π
n
{\displaystyle 2\cos {\frac {(n-2k)\pi }{n}}}
Le cas
n
=
2
n
′
{\displaystyle n=2n'}
n
′
{\displaystyle n'}
n
=
n
′
{\displaystyle n=n'}
Exemple : n = 40
modifier
Le polynôme minimal (sur
Q
{\displaystyle \mathbb {Q} }
2
cos
2
k
π
40
{\displaystyle 2\cos {\frac {2k\pi }{40}}}
k
=
1
,
3
,
7
,
9
{\displaystyle k=1,3,7,9}
P
40
(
X
)
=
P
20
(
X
2
−
2
)
=
P
10
(
(
X
2
−
2
)
2
−
2
)
=
P
5
(
2
−
(
X
2
−
2
)
2
)
=
P
5
(
−
X
4
+
4
X
2
−
2
)
=
(
−
X
4
+
4
X
2
−
2
)
2
+
(
−
X
4
+
4
X
2
−
2
)
−
1
=
X
8
−
8
X
6
+
19
X
4
−
12
X
2
+
1.
{\displaystyle {\begin{aligned}P_{40}(X)&=P_{20}(X^{2}-2)\\&=P_{10}((X^{2}-2)^{2}-2)\\&=P_{5}(2-(X^{2}-2)^{2})\\&=P_{5}(-X^{4}+4X^{2}-2)\\&=(-X^{4}+4X^{2}-2)^{2}+(-X^{4}+4X^{2}-2)-1\\&=X^{8}-8X^{6}+19X^{4}-12X^{2}+1.\end{aligned}}}
En posant
x
=
y
+
2
{\displaystyle x=y+2}
x
4
−
8
x
3
+
19
x
2
−
12
x
+
1
=
0
{\displaystyle x^{4}-8x^{3}+19x^{2}-12x+1=0}
y
4
−
5
y
2
+
5
=
0
{\displaystyle y^{4}-5y^{2}+5=0}
2
cos
π
20
=
2
+
5
+
5
2
,
2
cos
3
π
20
=
2
+
5
−
5
2
,
2
cos
7
π
20
=
2
−
5
−
5
2
,
2
cos
9
π
20
=
2
−
5
+
5
2
{\displaystyle 2\cos {\frac {\pi }{20}}={\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}},\quad 2\cos {\frac {3\pi }{20}}={\sqrt {2+{\sqrt {\frac {5-{\sqrt {5}}}{2}}}}},\quad 2\cos {\frac {7\pi }{20}}={\sqrt {2-{\sqrt {\frac {5-{\sqrt {5}}}{2}}}}},\quad 2\cos {\frac {9\pi }{20}}={\sqrt {2-{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}
soit
2
cos
π
20
=
1
+
5
+
a
−
b
2
2
,
2
cos
3
π
20
=
−
1
+
5
+
a
+
b
2
2
,
2
cos
7
π
20
=
1
−
5
+
a
+
b
2
2
,
2
cos
9
π
20
=
1
+
5
−
a
+
b
2
2
{\displaystyle 2\cos {\frac {\pi }{20}}={\frac {1+{\sqrt {5}}+a-b}{2{\sqrt {2}}}},\quad 2\cos {\frac {3\pi }{20}}={\frac {-1+{\sqrt {5}}+a+b}{2{\sqrt {2}}}},\quad 2\cos {\frac {7\pi }{20}}={\frac {1-{\sqrt {5}}+a+b}{2{\sqrt {2}}}},\quad 2\cos {\frac {9\pi }{20}}={\frac {1+{\sqrt {5}}-a+b}{2{\sqrt {2}}}}}
avec
a
=
5
+
2
5
,
b
=
5
−
2
5
{\displaystyle a={\sqrt {5+2{\sqrt {5}}}},\quad b={\sqrt {5-2{\sqrt {5}}}}}
Le groupe
(
Z
/
40
Z
)
/
{
1
,
−
1
}
{\displaystyle \left(\mathbb {Z} /40\mathbb {Z} \right)/\{1,-1\}}
φ
(
40
)
/
2
=
8
{\displaystyle \varphi (40)/2=8}
Le premier,
H
1
{\displaystyle H_{1}}
(
Z
/
40
Z
)
/
{
1
,
−
1
}
{\displaystyle \left(\mathbb {Z} /40\mathbb {Z} \right)/\{1,-1\}}
H
1
=
{
±
1
,
±
7
,
±
9
,
±
17
}
{\displaystyle H_{1}=\{\pm 1,\pm 7,\pm 9,\pm 17\}}
Avec les mêmes notations, les deux autres sont :
H
2
=
{
±
1
,
±
3
,
±
9
,
±
13
}
{\displaystyle H_{2}=\{\pm 1,\pm 3,\pm 9,\pm 13\}}
H
3
=
{
±
1
,
±
9
,
±
11
,
±
19
}
{\displaystyle H_{3}=\{\pm 1,\pm 9,\pm 11,\pm 19\}}
Ils fournissent trois factorisations,
P
40
(
X
)
=
Q
1
(
X
)
Q
1
(
−
X
)
=
Q
2
(
X
)
Q
2
(
−
X
)
=
Q
3
(
X
)
Q
4
(
X
)
{\displaystyle P_{40}(X)=Q_{1}(X)Q_{1}(-X)=Q_{2}(X)Q_{2}(-X)=Q_{3}(X)Q_{4}(X)}
avec
Q
1
(
X
)
=
(
X
−
2
cos
π
20
)
(
X
−
2
cos
7
π
20
)
(
X
−
2
cos
9
π
20
)
(
X
−
2
cos
17
π
20
)
=
(
X
−
1
+
5
+
a
−
b
2
2
)
(
X
−
1
−
5
+
a
+
b
2
2
)
(
X
−
1
+
5
−
a
+
b
2
2
)
(
X
+
−
1
+
5
+
a
+
b
2
2
)
=
X
4
−
X
3
2
−
3
X
2
+
3
X
2
−
1
{\displaystyle {\begin{aligned}Q_{1}(X)&=\left(X-2\cos {\frac {\pi }{20}}\right)\left(X-2\cos {\frac {7\pi }{20}}\right)\left(X-2\cos {\frac {9\pi }{20}}\right)\left(X-2\cos {\frac {17\pi }{20}}\right)\\&=\left(X-{\frac {1+{\sqrt {5}}+a-b}{2{\sqrt {2}}}}\right)\left(X-{\frac {1-{\sqrt {5}}+a+b}{2{\sqrt {2}}}}\right)\left(X-{\frac {1+{\sqrt {5}}-a+b}{2{\sqrt {2}}}}\right)\left(X+{\frac {-1+{\sqrt {5}}+a+b}{2{\sqrt {2}}}}\right)\\&=X^{4}-X^{3}{\sqrt {2}}-3X^{2}+3X{\sqrt {2}}-1\end{aligned}}}
et (par des calculs analogues) :
Q
2
(
X
)
=
X
4
−
10
X
3
+
X
2
+
10
X
−
1
,
Q
3
(
X
)
=
X
4
−
4
X
2
+
3
−
5
2
,
Q
4
(
X
)
=
X
4
−
4
X
2
+
3
+
5
2
{\displaystyle Q_{2}(X)=X^{4}-{\sqrt {10}}X^{3}+X^{2}+{\sqrt {10}}X-1,\quad Q_{3}(X)=X^{4}-4X^{2}+{\frac {3-{\sqrt {5}}}{2}},\quad Q_{4}(X)=X^{4}-4X^{2}+{\frac {3+{\sqrt {5}}}{2}}}
Nombres rationnels
modifier
φ
(
n
)
=
2
{\displaystyle \varphi (n)=2}
n
=
3
,
4
,
6
{\displaystyle n=3,4,6}
Les polynômes minimaux des nombres
2
cos
2
π
3
,
2
cos
π
2
{\displaystyle 2\cos {\frac {2\pi }{3}},\quad 2\cos {\frac {\pi }{2}}}
sont respectivement :
X
+
1
,
X
{\displaystyle X+1,\qquad X}
Irrationnels quadratiques
modifier
Nombres de degré 3
modifier
φ
(
n
)
=
6
{\displaystyle \varphi (n)=6}
n
=
7
,
9
,
14
,
18
{\displaystyle n=7,9,14,18}
X
3
+
X
2
−
2
X
−
1
{\displaystyle X^{3}+X^{2}-2X-1}
2
cos
2
π
7
,
2
cos
4
π
7
,
2
cos
6
π
7
{\displaystyle 2\cos {\frac {2\pi }{7}},\quad 2\cos {\frac {4\pi }{7}},\quad 2\cos {\frac {6\pi }{7}}}
X
3
−
3
X
+
1
{\displaystyle X^{3}-3X+1}
2
cos
2
π
9
,
2
cos
4
π
9
,
2
cos
8
π
9
{\displaystyle 2\cos {\frac {2\pi }{9}},\quad 2\cos {\frac {4\pi }{9}},\quad 2\cos {\frac {8\pi }{9}}}
Nombres de degré 4
modifier
φ
(
n
)
=
8
{\displaystyle \varphi (n)=8}
n
=
15
,
16
,
20
,
24
,
30
{\displaystyle n=15,16,20,24,30}
X
2
−
5
+
1
2
X
−
3
−
5
2
{\displaystyle X^{2}-{\frac {{\sqrt {5}}+1}{2}}X-{\frac {3-{\sqrt {5}}}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
2
π
15
,
2
cos
8
π
15
{\displaystyle 2\cos {\frac {2\pi }{15}},\quad 2\cos {\frac {8\pi }{15}}}
X
2
+
5
−
1
2
X
−
3
+
5
2
{\displaystyle X^{2}+{\frac {{\sqrt {5}}-1}{2}}X-{\frac {3+{\sqrt {5}}}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
4
π
15
,
2
cos
14
π
15
{\displaystyle 2\cos {\frac {4\pi }{15}},\quad 2\cos {\frac {14\pi }{15}}}
X
2
−
2
−
2
{\displaystyle X^{2}-2-{\sqrt {2}}}
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2
cos
π
8
{\displaystyle 2\cos {\frac {\pi }{8}}}
X
2
−
2
+
2
{\displaystyle X^{2}-2+{\sqrt {2}}}
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2
cos
3
π
8
{\displaystyle 2\cos {\frac {3\pi }{8}}}
X
2
−
5
+
5
2
{\displaystyle X^{2}-{\frac {5+{\sqrt {5}}}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
π
10
{\displaystyle 2\cos {\frac {\pi }{10}}}
X
2
−
5
−
5
2
{\displaystyle X^{2}-{\frac {5-{\sqrt {5}}}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
3
π
10
{\displaystyle 2\cos {\frac {3\pi }{10}}}
X
2
−
2
X
−
1
{\displaystyle X^{2}-{\sqrt {2}}X-1}
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2
cos
π
12
{\displaystyle 2\cos {\frac {\pi }{12}}}
2
cos
7
π
12
{\displaystyle 2\cos {\frac {7\pi }{12}}}
X
2
−
2
−
3
{\displaystyle X^{2}-2-{\sqrt {3}}}
Q
(
3
)
{\displaystyle \mathbb {Q} ({\sqrt {3}})}
2
cos
π
12
{\displaystyle 2\cos {\frac {\pi }{12}}}
X
2
−
6
X
+
1
{\displaystyle X^{2}-{\sqrt {6}}X+1}
Q
(
6
)
{\displaystyle \mathbb {Q} ({\sqrt {6}})}
2
cos
π
12
{\displaystyle 2\cos {\frac {\pi }{12}}}
2
cos
5
π
12
{\displaystyle 2\cos {\frac {5\pi }{12}}}
Nombres de degré 5
modifier
φ
(
n
)
=
10
{\displaystyle \varphi (n)=10}
n
=
11
,
22
{\displaystyle n=11,22}
X
5
+
X
4
−
4
X
3
−
3
X
2
+
3
X
+
1
{\displaystyle X^{5}+X^{4}-4X^{3}-3X^{2}+3X+1}
2
cos
2
k
π
11
,
k
=
1
,
2
,
3
,
4
,
5
{\displaystyle 2\cos {\frac {2^{k}\pi }{11}},\quad k=1,2,3,4,5}
Nombres de degré 6
modifier
φ
(
n
)
=
12
{\displaystyle \varphi (n)=12}
n
=
13
,
21
,
26
,
28
,
36
,
42
{\displaystyle n=13,21,26,28,36,42}
X
3
+
1
−
13
2
X
2
−
X
−
3
−
13
2
{\displaystyle X^{3}+{\frac {1-{\sqrt {13}}}{2}}X^{2}-X-{\frac {3-{\sqrt {13}}}{2}}}
Q
(
13
)
{\displaystyle \mathbb {Q} ({\sqrt {13}})}
2
cos
2
π
13
,
2
cos
6
π
13
,
2
cos
8
π
13
{\displaystyle 2\cos {\frac {2\pi }{13}},\quad 2\cos {\frac {6\pi }{13}},\quad 2\cos {\frac {8\pi }{13}}}
X
3
+
1
+
13
2
X
2
−
X
−
3
+
13
2
{\displaystyle X^{3}+{\frac {1+{\sqrt {13}}}{2}}X^{2}-X-{\frac {3+{\sqrt {13}}}{2}}}
Q
(
13
)
{\displaystyle \mathbb {Q} ({\sqrt {13}})}
2
cos
4
π
13
,
2
cos
10
π
13
,
2
cos
12
π
13
{\displaystyle 2\cos {\frac {4\pi }{13}},\quad 2\cos {\frac {10\pi }{13}},\quad 2\cos {\frac {12\pi }{13}}}
X
3
−
1
+
21
2
X
2
−
1
−
21
2
X
−
5
−
21
2
{\displaystyle X^{3}-{\frac {1+{\sqrt {21}}}{2}}X^{2}-{\frac {1-{\sqrt {21}}}{2}}X-{\frac {5-{\sqrt {21}}}{2}}}
Q
(
21
)
{\displaystyle \mathbb {Q} ({\sqrt {21}})}
2
cos
2
π
21
,
2
cos
8
π
21
,
2
cos
10
π
21
{\displaystyle 2\cos {\frac {2\pi }{21}},\quad 2\cos {\frac {8\pi }{21}},\quad 2\cos {\frac {10\pi }{21}}}
X
3
−
1
−
21
2
X
2
−
1
+
21
2
X
−
5
+
21
2
{\displaystyle X^{3}-{\frac {1-{\sqrt {21}}}{2}}X^{2}-{\frac {1+{\sqrt {21}}}{2}}X-{\frac {5+{\sqrt {21}}}{2}}}
Q
(
21
)
{\displaystyle \mathbb {Q} ({\sqrt {21}})}
2
cos
4
π
21
,
2
cos
16
π
21
,
2
cos
20
π
21
{\displaystyle 2\cos {\frac {4\pi }{21}},\quad 2\cos {\frac {16\pi }{21}},\quad 2\cos {\frac {20\pi }{21}}}
X
3
−
X
2
7
+
7
{\displaystyle X^{3}-X^{2}{\sqrt {7}}+{\sqrt {7}}}
Q
(
7
)
{\displaystyle \mathbb {Q} ({\sqrt {7}})}
2
cos
π
14
,
2
cos
3
π
14
,
2
cos
9
π
14
{\displaystyle 2\cos {\frac {\pi }{14}},\quad 2\cos {\frac {3\pi }{14}},\quad 2\cos {\frac {9\pi }{14}}}
X
3
−
3
X
−
3
{\displaystyle X^{3}-3X-{\sqrt {3}}}
Q
(
3
)
{\displaystyle \mathbb {Q} ({\sqrt {3}})}
2
cos
π
18
,
2
cos
11
π
18
,
2
cos
13
π
18
{\displaystyle 2\cos {\frac {\pi }{18}},\quad 2\cos {\frac {11\pi }{18}},\quad 2\cos {\frac {13\pi }{18}}}
Nombres de degré 8
modifier
φ
(
n
)
=
16
{\displaystyle \varphi (n)=16}
n
=
17
,
32
,
34
,
40
,
48
,
60
{\displaystyle n=17,32,34,40,48,60}
X
4
+
1
−
17
2
X
3
−
3
+
17
2
X
2
+
(
2
+
17
)
X
−
1
{\displaystyle X^{4}+{\frac {1-{\sqrt {17}}}{2}}X^{3}-{\frac {3+{\sqrt {17}}}{2}}X^{2}+(2+{\sqrt {17}})X-1}
Q
(
17
)
{\displaystyle \mathbb {Q} ({\sqrt {17}})}
2
cos
2
k
π
17
,
k
=
1
,
2
,
3
,
4
{\displaystyle 2\cos {\frac {2^{k}\pi }{17}},\quad k=1,2,3,4}
X
4
+
1
+
17
2
X
3
−
3
−
17
2
X
2
+
(
2
−
17
)
X
−
1
{\displaystyle X^{4}+{\frac {1+{\sqrt {17}}}{2}}X^{3}-{\frac {3-{\sqrt {17}}}{2}}X^{2}+(2-{\sqrt {17}})X-1}
Q
(
17
)
{\displaystyle \mathbb {Q} ({\sqrt {17}})}
2
cos
2
k
×
3
π
17
,
k
=
1
,
2
,
3
,
4
{\displaystyle 2\cos {\frac {2^{k}\times 3\pi }{17}},\quad k=1,2,3,4}
X
4
−
4
X
2
+
2
−
2
{\displaystyle X^{4}-4X^{2}+2-{\sqrt {2}}}
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2
cos
π
16
,
2
cos
7
π
16
{\displaystyle 2\cos {\frac {\pi }{16}},\quad 2\cos {\frac {7\pi }{16}}}
X
4
−
4
X
2
+
2
+
2
{\displaystyle X^{4}-4X^{2}+2+{\sqrt {2}}}
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2
cos
3
π
16
,
2
cos
5
π
16
{\displaystyle 2\cos {\frac {3\pi }{16}},\quad 2\cos {\frac {5\pi }{16}}}
X
4
−
X
3
2
−
3
X
2
+
3
X
2
−
1
{\displaystyle X^{4}-X^{3}{\sqrt {2}}-3X^{2}+3X{\sqrt {2}}-1}
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2
cos
7
k
π
20
,
k
=
0
,
1
,
2
,
3
{\displaystyle 2\cos {\frac {7^{k}\pi }{20}},\quad k=0,1,2,3}
X
4
−
10
X
3
+
X
2
+
10
X
−
1
{\displaystyle X^{4}-{\sqrt {10}}X^{3}+X^{2}+{\sqrt {10}}X-1}
Q
(
10
)
{\displaystyle \mathbb {Q} ({\sqrt {10}})}
2
cos
3
k
π
20
,
k
=
0
,
1
,
2
,
3
{\displaystyle 2\cos {\frac {3^{k}\pi }{20}},\quad k=0,1,2,3}
X
4
−
4
X
2
+
3
−
5
2
{\displaystyle X^{4}-4X^{2}+{\frac {3-{\sqrt {5}}}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
π
20
,
2
cos
9
π
20
{\displaystyle 2\cos {\frac {\pi }{20}},\quad 2\cos {\frac {9\pi }{20}}}
X
4
−
4
X
2
+
3
+
5
2
{\displaystyle X^{4}-4X^{2}+{\frac {3+{\sqrt {5}}}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
3
π
20
,
2
cos
7
π
20
{\displaystyle 2\cos {\frac {3\pi }{20}},\quad 2\cos {\frac {7\pi }{20}}}
X
4
−
4
X
2
+
2
−
3
{\displaystyle X^{4}-4X^{2}+2-{\sqrt {3}}}
Q
(
3
)
{\displaystyle \mathbb {Q} ({\sqrt {3}})}
2
cos
π
24
,
2
cos
11
π
24
{\displaystyle 2\cos {\frac {\pi }{24}},\quad 2\cos {\frac {11\pi }{24}}}
X
4
−
4
X
2
+
2
+
3
{\displaystyle X^{4}-4X^{2}+2+{\sqrt {3}}}
Q
(
3
)
{\displaystyle \mathbb {Q} ({\sqrt {3}})}
2
cos
5
π
24
,
2
cos
7
π
24
{\displaystyle 2\cos {\frac {5\pi }{24}},\quad 2\cos {\frac {7\pi }{24}}}
X
4
−
X
3
15
+
4
X
2
−
1
{\displaystyle X^{4}-X^{3}{\sqrt {15}}+4X^{2}-1}
Q
(
15
)
{\displaystyle \mathbb {Q} ({\sqrt {15}})}
2
cos
7
k
π
30
,
k
=
0
,
1
,
2
,
3
{\displaystyle 2\cos {\frac {7^{k}\pi }{30}},\quad k=0,1,2,3}
Nombres de degré 10
modifier
φ
(
n
)
=
20
{\displaystyle \varphi (n)=20}
n
=
25
,
33
,
44
,
50
,
66
{\displaystyle n=25,33,44,50,66}
X
5
−
5
X
3
+
5
X
−
5
2
+
1
2
{\displaystyle X^{5}-5X^{3}+5X-{\frac {\sqrt {5}}{2}}+{\frac {1}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
4
k
×
2
π
25
,
k
=
0
,
1
,
2
,
3
,
4
{\displaystyle 2\cos {\frac {4^{k}\times 2\pi }{25}},\quad k=0,1,2,3,4}
X
5
−
5
X
3
+
5
X
+
5
2
+
1
2
{\displaystyle X^{5}-5X^{3}+5X+{\frac {\sqrt {5}}{2}}+{\frac {1}{2}}}
Q
(
5
)
{\displaystyle \mathbb {Q} ({\sqrt {5}})}
2
cos
4
k
×
π
25
,
k
=
1
,
2
,
3
,
4
,
5
{\displaystyle 2\cos {\frac {4^{k}\times \pi }{25}},\quad k=1,2,3,4,5}
X
10
−
X
9
−
10
X
8
+
10
X
7
+
34
X
6
−
34
X
5
−
43
X
4
+
43
X
3
+
12
X
2
−
12
X
+
1
{\displaystyle X^{10}-X^{9}-10X^{8}+10X^{7}+34X^{6}-34X^{5}-43X^{4}+43X^{3}+12X^{2}-12X+1}
2
cos
5
k
×
2
π
33
,
k
=
0
,
1
,
…
,
9
{\displaystyle 2\cos {\frac {5^{k}\times 2\pi }{33}},\quad k=0,1,\dots ,9}
Q
(
33
)
{\displaystyle \mathbb {Q} ({\sqrt {33}})}
X
5
−
X
4
11
+
3
X
2
11
−
11
X
+
11
{\displaystyle X^{5}-X^{4}{\sqrt {11}}+3X^{2}{\sqrt {11}}-11X+{\sqrt {11}}}
Q
(
11
)
{\displaystyle \mathbb {Q} ({\sqrt {11}})}
2
cos
5
k
π
22
,
k
=
0
,
1
,
2
,
3
,
4
{\displaystyle 2\cos {\frac {5^{k}\pi }{22}},\quad k=0,1,2,3,4}
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