Ci-dessous se trouvent, dans l'ordre des degrés croissants, les premiers polynômes minimaux des nombres de la forme 2.cos(k
π
{\displaystyle \pi }
π
{\displaystyle \pi }
x
−
1
x
+
1
x
−
2
x
+
2
{\displaystyle x-1\qquad x+1\qquad x-2\qquad x+2}
2
cos
(
π
3
)
,
2
cos
(
2
π
3
)
,
2
cos
(
0
)
,
2
cos
(
π
)
,
{\displaystyle 2\cos({\frac {\pi }{3}}),\qquad 2\cos({\frac {2\pi }{3}}),\qquad 2\cos(0),\qquad 2\cos(\pi ),~}
2
sin
(
π
6
)
,
2
sin
(
−
π
6
)
,
2
sin
(
π
2
)
,
2
sin
(
−
π
2
)
.
{\displaystyle 2\sin({\frac {\pi }{6}}),\qquad 2\sin({\frac {-\pi }{6}}),\qquad 2\sin({\frac {\pi }{2}}),\qquad 2\sin({\frac {-\pi }{2}}).~}
x
−
2
x
+
2
x
−
3
x
+
3
{\displaystyle x-{\sqrt {2}}\qquad x+{\sqrt {2}}\qquad x-{\sqrt {3}}\qquad x+{\sqrt {3}}}
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
2
cos
(
π
4
)
,
2
cos
(
3
π
4
)
,
2
cos
(
π
6
)
,
2
cos
(
5
π
6
)
,
{\displaystyle 2\cos({\frac {\pi }{4}}),\qquad 2\cos({\frac {3\pi }{4}}),\qquad 2\cos({\frac {\pi }{6}}),\qquad 2\cos({\frac {5\pi }{6}}),~}
2
sin
(
π
4
)
,
2
sin
(
−
π
4
)
,
2
sin
(
π
3
)
,
2
sin
(
−
π
3
)
.
{\displaystyle 2\sin({\frac {\pi }{4}}),\qquad 2\sin({\frac {-\pi }{4}}),\qquad 2\sin({\frac {\pi }{3}}),\qquad 2\sin({\frac {-\pi }{3}}).~}
x
2
−
x
2
−
1
{\displaystyle x^{2}-x{\sqrt {2}}-1~}
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
2
cos
(
π
12
)
,
2
cos
(
7
π
12
)
,
{\displaystyle 2\cos({\frac {\pi }{12}}),\qquad 2\cos({\frac {7\pi }{12}}),~}
2
sin
(
−
π
12
)
,
2
sin
(
5
π
12
)
.
{\displaystyle 2\sin({\frac {-\pi }{12}}),\qquad 2\sin({\frac {5\pi }{12}}).~}
x
2
+
x
2
−
1
{\displaystyle x^{2}+x{\sqrt {2}}-1~}
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
2
cos
(
5
π
12
)
,
2
cos
(
11
π
12
)
,
{\displaystyle 2\cos({\frac {5\pi }{12}}),\qquad 2\cos({\frac {11\pi }{12}}),~}
2
sin
(
π
12
)
,
2
sin
(
−
5
π
12
)
.
{\displaystyle 2\sin({\frac {\pi }{12}}),\qquad 2\sin({\frac {-5\pi }{12}}).~}
x
2
−
x
5
+
1
{\displaystyle x^{2}-x{\sqrt {5}}+1~}
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
π
5
)
,
2
cos
(
2
π
5
)
,
{\displaystyle 2\cos({\frac {\pi }{5}}),\qquad 2\cos({\frac {2\pi }{5}}),~}
2
sin
(
π
10
)
,
2
sin
(
3
π
10
)
.
{\displaystyle 2\sin({\frac {\pi }{10}}),\qquad 2\sin({\frac {3\pi }{10}}).~}
x
2
+
x
5
+
1
{\displaystyle x^{2}+x{\sqrt {5}}+1~}
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
3
π
5
)
,
2
cos
(
4
π
5
)
,
{\displaystyle 2\cos({\frac {3\pi }{5}}),\qquad 2\cos({\frac {4\pi }{5}}),~}
2
sin
(
−
π
10
)
,
2
sin
(
−
3
π
10
)
.
{\displaystyle 2\sin({\frac {-\pi }{10}}),\qquad 2\sin({\frac {-3\pi }{10}}).~}
x
3
−
x
2
−
2
x
+
1
{\displaystyle x^{3}-x^{2}-2x+1~}
2
cos
(
π
7
)
,
2
cos
(
3
π
7
)
,
2
cos
(
5
π
7
)
,
{\displaystyle 2\cos({\frac {\pi }{7}}),\qquad 2\cos({\frac {3\pi }{7}}),\qquad 2\cos({\frac {5\pi }{7}}),~}
2
sin
(
π
14
)
,
2
sin
(
−
3
π
14
)
,
2
sin
(
5
π
14
)
.
{\displaystyle 2\sin({\frac {\pi }{14}}),\qquad 2\sin({\frac {-3\pi }{14}}),\qquad 2\sin({\frac {5\pi }{14}}).~}
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}}),\qquad 2\cos({\frac {4\pi }{7}}),\qquad 2\cos({\frac {6\pi }{7}}),~}
2
sin
(
−
π
14
)
,
2
sin
(
3
π
14
)
,
2
sin
(
−
5
π
14
)
.
{\displaystyle 2\sin({\frac {-\pi }{14}}),\qquad 2\sin({\frac {3\pi }{14}}),\qquad 2\sin({\frac {-5\pi }{14}}).~}
x
3
−
3
x
−
1
{\displaystyle x^{3}-3x-1~}
2
cos
(
π
9
)
,
2
cos
(
5
π
9
)
,
2
cos
(
7
π
9
)
,
{\displaystyle 2\cos({\frac {\pi }{9}}),\qquad 2\cos({\frac {5\pi }{9}}),\qquad 2\cos({\frac {7\pi }{9}}),~}
2
sin
(
−
π
18
)
,
2
sin
(
−
5
π
18
)
,
2
sin
(
7
π
18
)
.
{\displaystyle 2\sin({\frac {-\pi }{18}}),\qquad 2\sin({\frac {-5\pi }{18}}),\qquad 2\sin({\frac {7\pi }{18}}).~}
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}}),\qquad 2\cos({\frac {4\pi }{9}}),\qquad 2\cos({\frac {8\pi }{9}}),~}
2
sin
(
π
18
)
,
2
sin
(
5
π
18
)
,
2
sin
(
−
7
π
18
)
.
{\displaystyle 2\sin({\frac {\pi }{18}}),\qquad 2\sin({\frac {5\pi }{18}}),\qquad 2\sin({\frac {-7\pi }{18}}).~}
x
3
−
x
2
7
+
7
{\displaystyle x^{3}-x^{2}{\sqrt {7}}+{\sqrt {7}}~}
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
2
cos
(
π
14
)
,
2
cos
(
3
π
14
)
,
2
cos
(
9
π
14
)
,
{\displaystyle 2\cos({\frac {\pi }{14}}),\qquad 2\cos({\frac {3\pi }{14}}),\qquad 2\cos({\frac {9\pi }{14}}),~}
2
sin
(
−
π
7
)
,
2
sin
(
2
π
7
)
,
2
sin
(
3
π
7
)
.
{\displaystyle 2\sin({\frac {-\pi }{7}}),\qquad 2\sin({\frac {2\pi }{7}}),\qquad 2\sin({\frac {3\pi }{7}}).~}
x
3
+
x
2
7
−
7
{\displaystyle x^{3}+x^{2}{\sqrt {7}}-{\sqrt {7}}~}
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
2
cos
(
5
π
14
)
,
2
cos
(
11
π
14
)
,
2
cos
(
13
π
14
)
,
{\displaystyle 2\cos({\frac {5\pi }{14}}),\qquad 2\cos({\frac {11\pi }{14}}),\qquad 2\cos({\frac {13\pi }{14}}),~}
2
sin
(
π
7
)
,
2
sin
(
−
2
π
7
)
,
2
sin
(
−
3
π
7
)
.
{\displaystyle 2\sin({\frac {\pi }{7}}),\qquad 2\sin({\frac {-2\pi }{7}}),\qquad 2\sin({\frac {-3\pi }{7}}).~}
x
3
−
3
x
−
3
{\displaystyle x^{3}-3x-{\sqrt {3}}~}
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
2
cos
(
π
18
)
,
2
cos
(
11
π
18
)
,
2
cos
(
13
π
18
)
,
{\displaystyle 2\cos({\frac {\pi }{18}}),\qquad 2\cos({\frac {11\pi }{18}}),\qquad 2\cos({\frac {13\pi }{18}}),~}
2
sin
(
−
π
9
)
,
2
sin
(
−
2
π
9
)
,
2
sin
(
4
π
9
)
.
{\displaystyle 2\sin({\frac {-\pi }{9}}),\qquad 2\sin({\frac {-2\pi }{9}}),\qquad 2\sin({\frac {4\pi }{9}}).~}
x
3
−
3
x
+
3
{\displaystyle x^{3}-3x+{\sqrt {3}}~}
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
2
cos
(
5
π
18
)
,
2
cos
(
7
π
18
)
,
2
cos
(
17
π
18
)
,
{\displaystyle 2\cos({\frac {5\pi }{18}}),\qquad 2\cos({\frac {7\pi }{18}}),\qquad 2\cos({\frac {17\pi }{18}}),~}
2
sin
(
π
9
)
,
2
sin
(
2
π
9
)
,
2
sin
(
−
4
π
9
)
.
{\displaystyle 2\sin({\frac {\pi }{9}}),\qquad 2\sin({\frac {2\pi }{9}}),\qquad 2\sin({\frac {-4\pi }{9}}).~}
2
x
3
+
(
1
+
13
)
x
2
−
2
x
−
3
−
13
{\displaystyle 2x^{3}+(1+{\sqrt {13}})x^{2}-2x-3-{\sqrt {13}}~}
Q
(
13
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {13}})[X]}
2
cos
(
4
π
13
)
,
2
cos
(
10
π
13
)
,
2
cos
(
12
π
13
)
,
{\displaystyle 2\cos({\frac {4\pi }{13}}),\qquad 2\cos({\frac {10\pi }{13}}),\qquad 2\cos({\frac {12\pi }{13}}),~}
2
sin
(
5
π
26
)
,
2
sin
(
−
7
π
26
)
,
2
sin
(
−
11
π
26
)
.
{\displaystyle 2\sin({\frac {5\pi }{26}}),\qquad 2\sin({\frac {-7\pi }{26}}),\qquad 2\sin({\frac {-11\pi }{26}}).~}
2
x
3
+
(
1
−
13
)
x
2
−
2
x
−
3
+
13
{\displaystyle 2x^{3}+(1-{\sqrt {13}})x^{2}-2x-3+{\sqrt {13}}~}
Q
(
13
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {13}})[X]}
2
cos
(
2
π
13
)
,
2
cos
(
6
π
13
)
,
2
cos
(
8
π
13
)
,
{\displaystyle 2\cos({\frac {2\pi }{13}}),\qquad 2\cos({\frac {6\pi }{13}}),\qquad 2\cos({\frac {8\pi }{13}}),~}
2
sin
(
π
26
)
,
2
sin
(
9
π
26
)
,
2
sin
(
−
3
π
26
)
.
{\displaystyle 2\sin({\frac {\pi }{26}}),\qquad 2\sin({\frac {9\pi }{26}}),\qquad 2\sin({\frac {-3\pi }{26}}).~}
2
x
3
−
(
1
−
13
)
x
2
−
2
x
+
3
−
13
{\displaystyle 2x^{3}-(1-{\sqrt {13}})x^{2}-2x+3-{\sqrt {13}}~}
Q
(
13
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {13}})[X]}
2
cos
(
5
π
13
)
,
2
cos
(
7
π
13
)
,
2
cos
(
11
π
13
)
,
{\displaystyle 2\cos({\frac {5\pi }{13}}),\qquad 2\cos({\frac {7\pi }{13}}),\qquad 2\cos({\frac {11\pi }{13}}),~}
2
sin
(
3
π
26
)
,
2
sin
(
−
π
26
)
,
2
sin
(
−
9
π
26
)
.
{\displaystyle 2\sin({\frac {3\pi }{26}}),\qquad 2\sin({\frac {-\pi }{26}}),\qquad 2\sin({\frac {-9\pi }{26}}).~}
2
x
3
−
(
1
+
13
)
x
2
−
2
x
+
3
+
13
{\displaystyle 2x^{3}-(1+{\sqrt {13}})x^{2}-2x+3+{\sqrt {13}}~}
Q
(
13
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {13}})[X]}
2
cos
(
π
13
)
,
2
cos
(
3
π
13
)
,
2
cos
(
9
π
13
)
,
{\displaystyle 2\cos({\frac {\pi }{13}}),\qquad 2\cos({\frac {3\pi }{13}}),\qquad 2\cos({\frac {9\pi }{13}}),~}
2
sin
(
7
π
26
)
,
2
sin
(
11
π
26
)
,
2
sin
(
−
5
π
26
)
.
{\displaystyle 2\sin({\frac {7\pi }{26}}),\qquad 2\sin({\frac {11\pi }{26}}),\qquad 2\sin({\frac {-5\pi }{26}}).~}
2
x
3
+
(
1
−
21
)
x
2
−
(
1
+
21
)
x
+
5
+
21
{\displaystyle 2x^{3}+(1-{\sqrt {21}})x^{2}-(1+{\sqrt {21}})x+5+{\sqrt {21}}~}
Q
(
21
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {21}})[X]}
2
cos
(
π
21
)
,
2
cos
(
5
π
21
)
,
2
cos
(
17
π
21
)
.
{\displaystyle 2\cos({\frac {\pi }{21}}),\qquad 2\cos({\frac {5\pi }{21}}),\qquad 2\cos({\frac {17\pi }{21}}).~}
2
x
3
+
(
1
+
21
)
x
2
−
(
1
−
21
)
x
+
5
−
21
{\displaystyle 2x^{3}+(1+{\sqrt {21}})x^{2}-(1-{\sqrt {21}})x+5-{\sqrt {21}}~}
Q
(
21
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {21}})[X]}
2
cos
(
11
π
21
)
,
2
cos
(
13
π
21
)
,
2
cos
(
19
π
21
)
.
{\displaystyle 2\cos({\frac {11\pi }{21}}),\qquad 2\cos({\frac {13\pi }{21}}),\qquad 2\cos({\frac {19\pi }{21}}).~}
2
x
3
−
(
1
+
21
)
x
2
−
(
1
−
21
)
x
−
5
+
21
{\displaystyle 2x^{3}-(1+{\sqrt {21}})x^{2}-(1-{\sqrt {21}})x-5+{\sqrt {21}}~}
Q
(
21
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {21}})[X]}
2
cos
(
2
π
21
)
,
2
cos
(
8
π
21
)
,
2
cos
(
10
π
21
)
.
{\displaystyle 2\cos({\frac {2\pi }{21}}),\qquad 2\cos({\frac {8\pi }{21}}),\qquad 2\cos({\frac {10\pi }{21}}).~}
2
x
3
−
(
1
−
21
)
x
2
−
(
1
+
21
)
x
−
5
−
21
{\displaystyle 2x^{3}-(1-{\sqrt {21}})x^{2}-(1+{\sqrt {21}})x-5-{\sqrt {21}}~}
Q
(
21
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {21}})[X]}
2
cos
(
4
π
21
)
,
2
cos
(
16
π
21
)
,
2
cos
(
20
π
21
)
.
{\displaystyle 2\cos({\frac {4\pi }{21}}),\qquad 2\cos({\frac {16\pi }{21}}),\qquad 2\cos({\frac {20\pi }{21}}).~}
x
4
−
4
x
2
+
2
{\displaystyle x^{4}-4x^{2}+2~}
2
cos
(
π
8
)
,
2
cos
(
3
π
8
)
,
2
cos
(
5
π
8
)
,
2
cos
(
7
π
8
)
,
{\displaystyle 2\cos({\frac {\pi }{8}}),\qquad 2\cos({\frac {3\pi }{8}}),\qquad 2\cos({\frac {5\pi }{8}}),\qquad 2\cos({\frac {7\pi }{8}}),~}
2
sin
(
π
8
)
,
2
sin
(
−
π
8
)
,
2
sin
(
3
π
8
)
,
2
sin
(
−
3
π
8
)
.
{\displaystyle 2\sin({\frac {\pi }{8}}),\qquad 2\sin({\frac {-\pi }{8}}),\qquad 2\sin({\frac {3\pi }{8}}),\qquad 2\sin({\frac {-3\pi }{8}}).~}
x
4
−
4
x
2
+
2
−
2
{\displaystyle x^{4}-4x^{2}+2-{\sqrt {2}}~}
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
2
cos
(
π
16
)
,
2
cos
(
7
π
16
)
,
2
cos
(
9
π
16
)
,
2
cos
(
15
π
16
)
,
{\displaystyle 2\cos({\frac {\pi }{16}}),\qquad 2\cos({\frac {7\pi }{16}}),\qquad 2\cos({\frac {9\pi }{16}}),\qquad 2\cos({\frac {15\pi }{16}}),~}
2
sin
(
π
16
)
,
2
sin
(
7
π
16
)
,
2
sin
(
−
π
16
)
,
2
sin
(
−
7
π
16
)
.
{\displaystyle 2\sin({\frac {\pi }{16}}),\qquad 2\sin({\frac {7\pi }{16}}),\qquad 2\sin({\frac {-\pi }{16}}),\qquad 2\sin({\frac {-7\pi }{16}}).~}
x
4
−
4
x
2
+
2
+
2
{\displaystyle x^{4}-4x^{2}+2+{\sqrt {2}}~}
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
2
cos
(
3
π
16
)
,
2
cos
(
5
π
16
)
,
2
cos
(
11
π
16
)
,
2
cos
(
13
π
16
)
,
{\displaystyle 2\cos({\frac {3\pi }{16}}),\qquad 2\cos({\frac {5\pi }{16}}),\qquad 2\cos({\frac {11\pi }{16}}),\qquad 2\cos({\frac {13\pi }{16}}),~}
2
sin
(
3
π
16
)
,
2
sin
(
5
π
16
)
,
2
sin
(
−
3
π
16
)
,
2
sin
(
−
5
π
16
)
.
{\displaystyle 2\sin({\frac {3\pi }{16}}),\qquad 2\sin({\frac {5\pi }{16}}),\qquad 2\sin({\frac {-3\pi }{16}}),\qquad 2\sin({\frac {-5\pi }{16}}).~}
x
4
−
4
x
2
+
2
−
3
{\displaystyle x^{4}-4x^{2}+2-{\sqrt {3}}~}
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
2
cos
(
π
24
)
,
2
cos
(
11
π
24
)
,
2
cos
(
13
π
24
)
,
2
cos
(
23
π
24
)
,
{\displaystyle 2\cos({\frac {\pi }{24}}),\qquad 2\cos({\frac {11\pi }{24}}),\qquad 2\cos({\frac {13\pi }{24}}),\qquad 2\cos({\frac {23\pi }{24}}),~}
2
sin
(
π
24
)
,
2
sin
(
11
π
24
)
,
2
sin
(
−
π
24
)
,
2
sin
(
−
11
π
24
)
.
{\displaystyle 2\sin({\frac {\pi }{24}}),\qquad 2\sin({\frac {11\pi }{24}}),\qquad 2\sin({\frac {-\pi }{24}}),\qquad 2\sin({\frac {-11\pi }{24}}).~}
x
4
−
4
x
2
+
2
+
3
{\displaystyle x^{4}-4x^{2}+2+{\sqrt {3}}~}
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
2
cos
(
5
π
24
)
,
2
cos
(
7
π
24
)
,
2
cos
(
17
π
24
)
,
2
cos
(
19
π
24
)
,
{\displaystyle 2\cos({\frac {5\pi }{24}}),\qquad 2\cos({\frac {7\pi }{24}}),\qquad 2\cos({\frac {17\pi }{24}}),\qquad 2\cos({\frac {19\pi }{24}}),~}
2
sin
(
5
π
24
)
,
2
sin
(
7
π
24
)
,
2
sin
(
−
5
π
24
)
,
2
sin
(
−
7
π
24
)
.
{\displaystyle 2\sin({\frac {5\pi }{24}}),\qquad 2\sin({\frac {7\pi }{24}}),\qquad 2\sin({\frac {-5\pi }{24}}),\qquad 2\sin({\frac {-7\pi }{24}}).~}
x
4
−
5
x
2
+
5
{\displaystyle x^{4}-5x^{2}+5~}
2
cos
(
π
10
)
,
2
cos
(
3
π
10
)
,
2
cos
(
7
π
10
)
,
2
cos
(
9
π
10
)
,
{\displaystyle 2\cos({\frac {\pi }{10}}),\qquad 2\cos({\frac {3\pi }{10}}),\qquad 2\cos({\frac {7\pi }{10}}),\qquad 2\cos({\frac {9\pi }{10}}),~}
2
sin
(
π
5
)
,
2
sin
(
−
π
5
)
,
2
sin
(
2
π
5
)
,
2
sin
(
−
2
π
5
)
.
{\displaystyle 2\sin({\frac {\pi }{5}}),\qquad 2\sin({\frac {-\pi }{5}}),\qquad 2\sin({\frac {2\pi }{5}}),\qquad 2\sin({\frac {-2\pi }{5}}).~}
x
4
−
x
3
5
−
2
x
2
+
2
x
5
+
5
{\displaystyle x^{4}-x^{3}{\sqrt {5}}-2x^{2}+2x{\sqrt {5}}+5~}
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
π
15
)
,
2
cos
(
2
π
15
)
,
2
cos
(
8
π
15
)
,
2
cos
(
11
π
15
)
,
{\displaystyle 2\cos({\frac {\pi }{15}}),\qquad 2\cos({\frac {2\pi }{15}}),\qquad 2\cos({\frac {8\pi }{15}}),\qquad 2\cos({\frac {11\pi }{15}}),~}
2
sin
(
−
π
30
)
,
2
sin
(
−
7
π
30
)
,
2
sin
(
11
π
30
)
,
2
sin
(
13
π
30
)
.
{\displaystyle 2\sin({\frac {-\pi }{30}}),\qquad 2\sin({\frac {-7\pi }{30}}),\qquad 2\sin({\frac {11\pi }{30}}),\qquad 2\sin({\frac {13\pi }{30}}).~}
x
4
+
x
3
5
−
2
x
2
−
2
x
5
+
5
{\displaystyle x^{4}+x^{3}{\sqrt {5}}-2x^{2}-2x{\sqrt {5}}+5~}
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
4
π
15
)
,
2
cos
(
7
π
15
)
,
2
cos
(
13
π
15
)
,
2
cos
(
14
π
15
)
,
{\displaystyle 2\cos({\frac {4\pi }{15}}),\qquad 2\cos({\frac {7\pi }{15}}),\qquad 2\cos({\frac {13\pi }{15}}),\qquad 2\cos({\frac {14\pi }{15}}),~}
2
sin
(
π
30
)
,
2
sin
(
7
π
30
)
,
2
sin
(
−
11
π
30
)
,
2
sin
(
−
13
π
30
)
.
{\displaystyle 2\sin({\frac {\pi }{30}}),\qquad 2\sin({\frac {7\pi }{30}}),\qquad 2\sin({\frac {-11\pi }{30}}),\qquad 2\sin({\frac {-13\pi }{30}}).~}
2
x
4
−
(
1
−
17
)
x
3
−
(
3
+
17
)
x
2
−
2
(
2
+
17
)
x
−
2
{\displaystyle 2x^{4}-(1-{\sqrt {17}})x^{3}-(3+{\sqrt {17}})x^{2}-2(2+{\sqrt {17}})x-2~}
Q
(
17
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {17}})[X]}
2
cos
(
π
17
)
,
2
cos
(
9
π
17
)
,
2
cos
(
13
π
17
)
,
2
cos
(
15
π
17
)
,
{\displaystyle 2\cos({\frac {\pi }{17}}),\qquad 2\cos({\frac {9\pi }{17}}),\qquad 2\cos({\frac {13\pi }{17}}),\qquad 2\cos({\frac {15\pi }{17}}),~}
2
sin
(
15
π
34
)
,
2
sin
(
−
1
π
34
)
,
2
sin
(
−
9
π
34
)
,
2
sin
(
−
13
π
34
)
.
{\displaystyle 2\sin({\frac {15\pi }{34}}),\qquad 2\sin({\frac {-1\pi }{34}}),\qquad 2\sin({\frac {-9\pi }{34}}),\qquad 2\sin({\frac {-13\pi }{34}}).~}
2
x
4
−
(
1
+
17
)
x
3
−
(
3
−
17
)
x
2
−
2
(
2
−
17
)
x
−
2
{\displaystyle 2x^{4}-(1+{\sqrt {17}})x^{3}-(3-{\sqrt {17}})x^{2}-2(2-{\sqrt {17}})x-2~}
Q
(
17
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {17}})[X]}
2
cos
(
3
π
17
)
,
2
cos
(
5
π
17
)
,
2
cos
(
7
π
17
)
,
2
cos
(
11
π
17
)
,
{\displaystyle 2\cos({\frac {3\pi }{17}}),\qquad 2\cos({\frac {5\pi }{17}}),\qquad 2\cos({\frac {7\pi }{17}}),\qquad 2\cos({\frac {11\pi }{17}}),~}
2
sin
(
3
π
34
)
,
2
sin
(
7
π
34
)
,
2
sin
(
11
π
34
)
,
2
sin
(
−
5
π
34
)
.
{\displaystyle 2\sin({\frac {3\pi }{34}}),\qquad 2\sin({\frac {7\pi }{34}}),\qquad 2\sin({\frac {11\pi }{34}}),\qquad 2\sin({\frac {-5\pi }{34}}).~}
2
x
4
+
(
1
+
17
)
x
3
−
(
3
−
17
)
x
2
+
2
(
2
−
17
)
x
−
2
{\displaystyle 2x^{4}+(1+{\sqrt {17}})x^{3}-(3-{\sqrt {17}})x^{2}+2(2-{\sqrt {17}})x-2~}
Q
(
17
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {17}})[X]}
2
cos
(
6
π
17
)
,
2
cos
(
10
π
17
)
,
2
cos
(
12
π
17
)
,
2
cos
(
14
π
17
)
,
{\displaystyle 2\cos({\frac {6\pi }{17}}),\qquad 2\cos({\frac {10\pi }{17}}),\qquad 2\cos({\frac {12\pi }{17}}),\qquad 2\cos({\frac {14\pi }{17}}),~}
2
sin
(
5
π
34
)
,
2
sin
(
−
3
π
34
)
,
2
sin
(
−
7
π
34
)
,
2
sin
(
−
11
π
34
)
.
{\displaystyle 2\sin({\frac {5\pi }{34}}),\qquad 2\sin({\frac {-3\pi }{34}}),\qquad 2\sin({\frac {-7\pi }{34}}),\qquad 2\sin({\frac {-11\pi }{34}}).~}
2
x
4
+
(
1
−
17
)
x
3
−
(
3
+
17
)
x
2
+
2
(
2
+
17
)
x
−
2
{\displaystyle 2x^{4}+(1-{\sqrt {17}})x^{3}-(3+{\sqrt {17}})x^{2}+2(2+{\sqrt {17}})x-2~}
Q
(
17
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {17}})[X]}
2
cos
(
2
π
17
)
,
2
cos
(
4
π
17
)
,
2
cos
(
8
π
17
)
,
2
cos
(
16
π
17
)
,
{\displaystyle 2\cos({\frac {2\pi }{17}}),\qquad 2\cos({\frac {4\pi }{17}}),\qquad 2\cos({\frac {8\pi }{17}}),\qquad 2\cos({\frac {16\pi }{17}}),~}
2
sin
(
π
34
)
,
2
sin
(
9
π
34
)
,
2
sin
(
13
π
34
)
,
2
sin
(
−
15
π
34
)
.
{\displaystyle 2\sin({\frac {\pi }{34}}),\qquad 2\sin({\frac {9\pi }{34}}),\qquad 2\sin({\frac {13\pi }{34}}),\qquad 2\sin({\frac {-15\pi }{34}}).~}
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
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
2
cos
(
π
20
)
,
2
cos
(
7
π
20
)
,
2
cos
(
9
π
20
)
,
2
cos
(
17
π
20
)
,
{\displaystyle 2\cos({\frac {\pi }{20}}),\qquad 2\cos({\frac {7\pi }{20}}),\qquad 2\cos({\frac {9\pi }{20}}),\qquad 2\cos({\frac {17\pi }{20}}),~}
2
sin
(
π
20
)
,
2
sin
(
3
π
20
)
,
2
sin
(
−
7
π
20
)
,
2
sin
(
9
π
20
)
.
{\displaystyle 2\sin({\frac {\pi }{20}}),\qquad 2\sin({\frac {3\pi }{20}}),\qquad 2\sin({\frac {-7\pi }{20}}),\qquad 2\sin({\frac {9\pi }{20}}).~}
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
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
2
cos
(
3
π
20
)
,
2
cos
(
11
π
20
)
,
2
cos
(
13
π
20
)
,
2
cos
(
19
π
20
)
,
{\displaystyle 2\cos({\frac {3\pi }{20}}),\qquad 2\cos({\frac {11\pi }{20}}),\qquad 2\cos({\frac {13\pi }{20}}),\qquad 2\cos({\frac {19\pi }{20}}),~}
2
sin
(
−
π
20
)
,
2
sin
(
−
3
π
20
)
,
2
sin
(
7
π
20
)
,
2
sin
(
−
9
π
20
)
.
{\displaystyle 2\sin({\frac {-\pi }{20}}),\qquad 2\sin({\frac {-3\pi }{20}}),\qquad 2\sin({\frac {7\pi }{20}}),\qquad 2\sin({\frac {-9\pi }{20}}).~}
x
4
−
x
3
15
+
4
x
2
−
1
{\displaystyle x^{4}-x^{3}{\sqrt {15}}+4x^{2}-1~}
Q
(
15
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {15}})[X]}
2
cos
(
π
30
)
,
2
cos
(
7
π
30
)
,
2
cos
(
11
π
30
)
,
2
cos
(
17
π
30
)
,
{\displaystyle 2\cos({\frac {\pi }{30}}),\qquad 2\cos({\frac {7\pi }{30}}),\qquad 2\cos({\frac {11\pi }{30}}),\qquad 2\cos({\frac {17\pi }{30}}),~}
2
sin
(
−
π
15
)
,
2
sin
(
2
π
15
)
,
2
sin
(
4
π
15
)
,
2
sin
(
7
π
15
)
.
{\displaystyle 2\sin({\frac {-\pi }{15}}),\qquad 2\sin({\frac {2\pi }{15}}),\qquad 2\sin({\frac {4\pi }{15}}),\qquad 2\sin({\frac {7\pi }{15}}).~}
x
4
+
x
3
15
+
4
x
2
−
1
{\displaystyle x^{4}+x^{3}{\sqrt {15}}+4x^{2}-1~}
Q
(
15
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {15}})[X]}
2
cos
(
13
π
30
)
,
2
cos
(
19
π
30
)
,
2
cos
(
23
π
30
)
,
2
cos
(
29
π
30
)
,
{\displaystyle 2\cos({\frac {13\pi }{30}}),\qquad 2\cos({\frac {19\pi }{30}}),\qquad 2\cos({\frac {23\pi }{30}}),\qquad 2\cos({\frac {29\pi }{30}}),~}
2
sin
(
π
15
)
,
2
sin
(
−
2
π
15
)
,
2
sin
(
−
4
π
15
)
,
2
sin
(
−
7
π
15
)
.
{\displaystyle 2\sin({\frac {\pi }{15}}),\qquad 2\sin({\frac {-2\pi }{15}}),\qquad 2\sin({\frac {-4\pi }{15}}),\qquad 2\sin({\frac {-7\pi }{15}}).~}
x
5
+
x
4
−
4
x
3
−
3
x
2
+
3
x
+
1
{\displaystyle x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1~}
est le polynôme minimal des nombres :
2
cos
(
2
π
11
)
,
2
cos
(
4
π
11
)
,
2
cos
(
6
π
11
)
,
2
cos
(
8
π
11
)
,
2
cos
(
10
π
11
)
.
{\displaystyle 2\cos({\frac {2\pi }{11}}),\qquad 2\cos({\frac {4\pi }{11}}),\qquad 2\cos({\frac {6\pi }{11}}),\qquad 2\cos({\frac {8\pi }{11}}),\qquad 2\cos({\frac {10\pi }{11}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
−
π
22
)
,
2
sin
(
3
π
22
)
,
2
sin
(
−
5
π
22
)
,
2
sin
(
7
π
22
)
,
2
sin
(
−
9
π
22
)
.
{\displaystyle 2\sin({\frac {-\pi }{22}}),\qquad 2\sin({\frac {3\pi }{22}}),\qquad 2\sin({\frac {-5\pi }{22}}),\qquad 2\sin({\frac {7\pi }{22}}),\qquad 2\sin({\frac {-9\pi }{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~}
Est le polynôme minimal des nombres :
2
cos
(
π
11
)
,
2
cos
(
3
π
11
)
,
2
cos
(
5
π
11
)
,
2
cos
(
7
π
11
)
,
2
cos
(
9
π
11
)
.
{\displaystyle 2\cos({\frac {\pi }{11}}),\qquad 2\cos({\frac {3\pi }{11}}),\qquad 2\cos({\frac {5\pi }{11}}),\qquad 2\cos({\frac {7\pi }{11}}),\qquad 2\cos({\frac {9\pi }{11}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
π
22
)
,
2
sin
(
−
3
π
22
)
,
2
sin
(
5
π
22
)
,
2
sin
(
−
7
π
22
)
,
2
sin
(
9
π
22
)
.
{\displaystyle 2\sin({\frac {\pi }{22}}),\qquad 2\sin({\frac {-3\pi }{22}}),\qquad 2\sin({\frac {5\pi }{22}}),\qquad 2\sin({\frac {-7\pi }{22}}),\qquad 2\sin({\frac {9\pi }{22}}).~}
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}}~}
Est le polynôme minimal dans
Q
(
11
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {11}})[X]}
2
cos
(
3
π
22
)
,
2
cos
(
13
π
22
)
,
2
cos
(
15
π
22
)
,
2
cos
(
17
π
22
)
,
2
cos
(
21
π
22
)
.
{\displaystyle 2\cos({\frac {3\pi }{22}}),\qquad 2\cos({\frac {13\pi }{22}}),\qquad 2\cos({\frac {15\pi }{22}}),\qquad 2\cos({\frac {17\pi }{22}}),\qquad 2\cos({\frac {21\pi }{22}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
−
π
11
)
,
2
sin
(
−
2
π
11
)
,
2
sin
(
−
3
π
11
)
,
2
sin
(
4
π
11
)
,
2
sin
(
−
5
π
11
)
.
{\displaystyle 2\sin({\frac {-\pi }{11}}),\qquad 2\sin({\frac {-2\pi }{11}}),\qquad 2\sin({\frac {-3\pi }{11}}),\qquad 2\sin({\frac {4\pi }{11}}),\qquad 2\sin({\frac {-5\pi }{11}}).~}
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}}~}
Est le polynôme minimal dans
Q
(
11
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {11}})[X]}
2
cos
(
π
22
)
,
2
cos
(
5
π
22
)
,
2
cos
(
7
π
22
)
,
2
cos
(
9
π
22
)
,
2
cos
(
19
π
22
)
.
{\displaystyle 2\cos({\frac {\pi }{22}}),\qquad 2\cos({\frac {5\pi }{22}}),\qquad 2\cos({\frac {7\pi }{22}}),\qquad 2\cos({\frac {9\pi }{22}}),\qquad 2\cos({\frac {19\pi }{22}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
π
11
)
,
2
sin
(
2
π
11
)
,
2
sin
(
3
π
11
)
,
2
sin
(
−
4
π
11
)
,
2
sin
(
5
π
11
)
.
{\displaystyle 2\sin({\frac {\pi }{11}}),\qquad 2\sin({\frac {2\pi }{11}}),\qquad 2\sin({\frac {3\pi }{11}}),\qquad 2\sin({\frac {-4\pi }{11}}),\qquad 2\sin({\frac {5\pi }{11}}).~}
2
x
5
−
10
x
3
+
10
x
−
5
−
1
{\displaystyle 2x^{5}-10x^{3}+10x-{\sqrt {5}}-1~}
Est le polynôme minimal dans
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
π
25
)
,
2
cos
(
9
π
25
)
,
2
cos
(
11
π
25
)
,
2
cos
(
19
π
25
)
,
2
cos
(
21
π
25
)
.
{\displaystyle 2\cos({\frac {\pi }{25}}),\qquad 2\cos({\frac {9\pi }{25}}),\qquad 2\cos({\frac {11\pi }{25}}),\qquad 2\cos({\frac {19\pi }{25}}),\qquad 2\cos({\frac {21\pi }{25}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
23
π
50
)
,
2
sin
(
7
π
50
)
,
2
sin
(
3
π
50
)
,
2
sin
(
−
13
π
50
)
,
2
sin
(
−
17
π
50
)
.
{\displaystyle 2\sin({\frac {23\pi }{50}}),\qquad 2\sin({\frac {7\pi }{50}}),\qquad 2\sin({\frac {3\pi }{50}}),\qquad 2\sin({\frac {-13\pi }{50}}),\qquad 2\sin({\frac {-17\pi }{50}}).~}
2
x
5
−
10
x
3
+
10
x
−
5
+
1
{\displaystyle 2x^{5}-10x^{3}+10x-{\sqrt {5}}+1~}
Est le polynôme minimal dans
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
2
π
25
)
,
2
cos
(
8
π
25
)
,
2
cos
(
12
π
25
)
,
2
cos
(
18
π
25
)
,
2
cos
(
22
π
25
)
.
{\displaystyle 2\cos({\frac {2\pi }{25}}),\qquad 2\cos({\frac {8\pi }{25}}),\qquad 2\cos({\frac {12\pi }{25}}),\qquad 2\cos({\frac {18\pi }{25}}),\qquad 2\cos({\frac {22\pi }{25}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
21
π
50
)
,
2
sin
(
9
π
50
)
,
2
sin
(
π
50
)
,
2
sin
(
−
11
π
50
)
,
2
sin
(
−
19
π
50
)
.
{\displaystyle 2\sin({\frac {21\pi }{50}}),\qquad 2\sin({\frac {9\pi }{50}}),\qquad 2\sin({\frac {\pi }{50}}),\qquad 2\sin({\frac {-11\pi }{50}}),\qquad 2\sin({\frac {-19\pi }{50}}).~}
2
x
5
−
10
x
3
+
10
x
+
5
−
1
{\displaystyle 2x^{5}-10x^{3}+10x+{\sqrt {5}}-1~}
Est le polynôme minimal dans
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
3
π
25
)
,
2
cos
(
7
π
25
)
,
2
cos
(
13
π
25
)
,
2
cos
(
17
π
25
)
,
2
cos
(
23
π
25
)
.
{\displaystyle 2\cos({\frac {3\pi }{25}}),\qquad 2\cos({\frac {7\pi }{25}}),\qquad 2\cos({\frac {13\pi }{25}}),\qquad 2\cos({\frac {17\pi }{25}}),\qquad 2\cos({\frac {23\pi }{25}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
19
π
50
)
,
2
sin
(
11
π
50
)
,
2
sin
(
−
π
50
)
,
2
sin
(
−
9
π
50
)
,
2
sin
(
−
21
π
50
)
.
{\displaystyle 2\sin({\frac {19\pi }{50}}),\qquad 2\sin({\frac {11\pi }{50}}),\qquad 2\sin({\frac {-\pi }{50}}),\qquad 2\sin({\frac {-9\pi }{50}}),\qquad 2\sin({\frac {-21\pi }{50}}).~}
2
x
5
−
10
x
3
+
10
x
+
5
+
1
{\displaystyle 2x^{5}-10x^{3}+10x+{\sqrt {5}}+1~}
Est le polynôme minimal dans
Q
(
5
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {5}})[X]}
2
cos
(
4
π
25
)
,
2
cos
(
6
π
25
)
,
2
cos
(
14
π
25
)
,
2
cos
(
16
π
25
)
,
2
cos
(
24
π
25
)
.
{\displaystyle 2\cos({\frac {4\pi }{25}}),\qquad 2\cos({\frac {6\pi }{25}}),\qquad 2\cos({\frac {14\pi }{25}}),\qquad 2\cos({\frac {16\pi }{25}}),\qquad 2\cos({\frac {24\pi }{25}}).~}
Qui peuvent aussi se mettre sous la forme :
2
sin
(
17
π
50
)
,
2
sin
(
13
π
50
)
,
2
sin
(
−
3
π
50
)
,
2
sin
(
−
7
π
50
)
,
2
sin
(
−
23
π
50
)
.
{\displaystyle 2\sin({\frac {17\pi }{50}}),\qquad 2\sin({\frac {13\pi }{50}}),\qquad 2\sin({\frac {-3\pi }{50}}),\qquad 2\sin({\frac {-7\pi }{50}}),\qquad 2\sin({\frac {-23\pi }{50}}).~}
Etc.
Ci-dessous se trouvent, dans l'ordre des degrés croissants, les premiers polynômes minimaux des nombres de la forme tan(kπ/n).
x
−
1
x
+
1
{\displaystyle x-1\qquad x+1}
sont respectivement les polynômes minimaux des nombres :
tan
(
π
4
)
,
tan
(
3
π
4
)
.
{\displaystyle \tan({\frac {\pi }{4}}),\qquad \tan({\frac {3\pi }{4}}).~}
x
−
3
x
+
3
x
−
1
3
x
+
1
3
{\displaystyle x-{\sqrt {3}}\qquad x+{\sqrt {3}}\qquad x-{\frac {1}{\sqrt {3}}}\qquad x+{\frac {1}{\sqrt {3}}}}
sont respectivement les polynômes minimaux dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
π
3
)
,
tan
(
2
π
3
)
,
tan
(
π
6
)
,
tan
(
5
π
6
)
.
{\displaystyle \tan({\frac {\pi }{3}}),\qquad \tan({\frac {2\pi }{3}}),\qquad \tan({\frac {\pi }{6}}),\qquad \tan({\frac {5\pi }{6}}).~}
x
2
−
4
x
+
1
{\displaystyle x^{2}-4x+1~}
Est le polynôme minimal des nombres :
tan
(
π
12
)
,
tan
(
5
π
12
)
.
{\displaystyle \tan({\frac {\pi }{12}}),\qquad \tan({\frac {5\pi }{12}}).~}
x
2
+
4
x
+
1
{\displaystyle x^{2}+4x+1~}
est le polynôme minimal des nombres :
tan
(
7
π
12
)
,
tan
(
11
π
12
)
.
{\displaystyle \tan({\frac {7\pi }{12}}),\qquad \tan({\frac {11\pi }{12}}).~}
x
2
−
2
x
2
+
1
{\displaystyle x^{2}-2x{\sqrt {2}}+1~}
est le polynôme minimal dans
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
tan
(
π
8
)
,
tan
(
3
π
8
)
.
{\displaystyle \tan({\frac {\pi }{8}}),\qquad \tan({\frac {3\pi }{8}}).~}
x
2
+
2
x
2
+
1
{\displaystyle x^{2}+2x{\sqrt {2}}+1~}
est le polynôme minimal dans
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
tan
(
5
π
8
)
,
tan
(
7
π
8
)
.
{\displaystyle \tan({\frac {5\pi }{8}}),\qquad \tan({\frac {7\pi }{8}}).~}
x
3
+
x
2
7
−
7
x
+
7
{\displaystyle x^{3}+x^{2}{\sqrt {7}}-7x+{\sqrt {7}}~}
est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
π
7
)
,
tan
(
2
π
7
)
,
tan
(
4
π
7
)
.
{\displaystyle \tan({\frac {\pi }{7}}),\qquad \tan({\frac {2\pi }{7}}),\qquad \tan({\frac {4\pi }{7}}).~}
x
3
−
x
2
7
−
7
x
−
7
{\displaystyle x^{3}-x^{2}{\sqrt {7}}-7x-{\sqrt {7}}~}
est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
3
π
7
)
,
tan
(
5
π
7
)
,
tan
(
6
π
7
)
.
{\displaystyle \tan({\frac {3\pi }{7}}),\qquad \tan({\frac {5\pi }{7}}),\qquad \tan({\frac {6\pi }{7}}).~}
x
3
−
3
x
2
3
−
3
x
+
3
{\displaystyle x^{3}-3x^{2}{\sqrt {3}}-3x+{\sqrt {3}}~}
est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
π
9
)
,
tan
(
4
π
9
)
,
tan
(
7
π
9
)
.
{\displaystyle \tan({\frac {\pi }{9}}),\qquad \tan({\frac {4\pi }{9}}),\qquad \tan({\frac {7\pi }{9}}).~}
x
3
+
3
x
2
3
−
3
x
−
3
{\displaystyle x^{3}+3x^{2}{\sqrt {3}}-3x-{\sqrt {3}}~}
est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
2
π
9
)
,
tan
(
5
π
9
)
,
tan
(
8
π
9
)
.
{\displaystyle \tan({\frac {2\pi }{9}}),\qquad \tan({\frac {5\pi }{9}}),\qquad \tan({\frac {8\pi }{9}}).~}
x
3
+
x
2
7
+
x
−
1
7
{\displaystyle x^{3}+x^{2}{\sqrt {7}}+x-{\frac {1}{\sqrt {7}}}~}
est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
π
14
)
,
tan
(
9
π
14
)
,
tan
(
11
π
14
)
.
{\displaystyle \tan({\frac {\pi }{14}}),\qquad \tan({\frac {9\pi }{14}}),\qquad \tan({\frac {11\pi }{14}}).~}
x
3
−
x
2
7
+
x
+
1
7
{\displaystyle x^{3}-x^{2}{\sqrt {7}}+x+{\frac {1}{\sqrt {7}}}~}
Est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
3
π
14
)
,
tan
(
5
π
14
)
,
tan
(
13
π
14
)
.
{\displaystyle \tan({\frac {3\pi }{14}}),\qquad \tan({\frac {5\pi }{14}}),\qquad \tan({\frac {13\pi }{14}}).~}
x
3
−
x
2
3
−
3
x
+
1
3
{\displaystyle x^{3}-x^{2}{\sqrt {3}}-3x+{\frac {1}{\sqrt {3}}}~}
est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
π
18
)
,
tan
(
13
π
18
)
,
tan
(
7
π
18
)
.
{\displaystyle \tan({\frac {\pi }{18}}),\qquad \tan({\frac {13\pi }{18}}),\qquad \tan({\frac {7\pi }{18}}).~}
x
3
+
x
2
3
−
3
x
−
1
3
{\displaystyle x^{3}+x^{2}{\sqrt {3}}-3x-{\frac {1}{\sqrt {3}}}~}
est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
5
π
18
)
,
tan
(
11
π
18
)
,
tan
(
17
π
18
)
.
{\displaystyle \tan({\frac {5\pi }{18}}),\qquad \tan({\frac {11\pi }{18}}),\qquad \tan({\frac {17\pi }{18}}).~}
(
2
7
+
3
3
)
x
3
+
x
2
−
(
2
7
+
3
)
x
+
1
{\displaystyle (2{\sqrt {7}}+3{\sqrt {3}})x^{3}+x^{2}-(2{\sqrt {7}}+{\sqrt {3}})x+1~}
est le polynôme minimal dans
Q
(
3
)
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})({\sqrt {7}})[X]}
tan
(
π
21
)
,
tan
(
4
π
21
)
,
tan
(
16
π
21
)
.
{\displaystyle \tan({\frac {\pi }{21}}),\qquad \tan({\frac {4\pi }{21}}),\qquad \tan({\frac {16\pi }{21}}).~}
(
2
7
−
3
3
)
x
3
+
x
2
−
(
2
7
−
3
)
x
+
1
{\displaystyle (2{\sqrt {7}}-3{\sqrt {3}})x^{3}+x^{2}-(2{\sqrt {7}}-{\sqrt {3}})x+1~}
est le polynôme minimal dans
Q
(
3
)
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})({\sqrt {7}})[X]}
tan
(
2
π
21
)
,
tan
(
8
π
21
)
,
tan
(
11
π
21
)
.
{\displaystyle \tan({\frac {2\pi }{21}}),\qquad \tan({\frac {8\pi }{21}}),\qquad \tan({\frac {11\pi }{21}}).~}
(
2
7
−
3
3
)
x
3
−
x
2
−
(
2
7
−
3
)
x
−
1
{\displaystyle (2{\sqrt {7}}-3{\sqrt {3}})x^{3}-x^{2}-(2{\sqrt {7}}-{\sqrt {3}})x-1~}
Est le polynôme minimal dans
Q
(
3
)
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})({\sqrt {7}})[X]}
tan
(
10
π
21
)
,
tan
(
13
π
21
)
,
tan
(
19
π
21
)
.
{\displaystyle \tan({\frac {10\pi }{21}}),\qquad \tan({\frac {13\pi }{21}}),\qquad \tan({\frac {19\pi }{21}}).~}
(
2
7
+
3
3
)
x
3
−
x
2
−
(
2
7
+
3
)
x
−
1
{\displaystyle (2{\sqrt {7}}+3{\sqrt {3}})x^{3}-x^{2}-(2{\sqrt {7}}+{\sqrt {3}})x-1~}
Est le polynôme minimal dans
Q
(
3
)
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})({\sqrt {7}})[X]}
tan
(
5
π
21
)
,
tan
(
17
π
21
)
,
tan
(
26
π
21
)
.
{\displaystyle \tan({\frac {5\pi }{21}}),\qquad \tan({\frac {17\pi }{21}}),\qquad \tan({\frac {26\pi }{21}}).~}
x
3
−
(
4
−
7
)
x
2
−
(
11
−
4
7
)
x
+
8
−
3
7
{\displaystyle x^{3}-(4-{\sqrt {7}})x^{2}-(11-4{\sqrt {7}})x+8-3{\sqrt {7}}~}
Est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
π
28
)
,
tan
(
9
π
28
)
,
tan
(
25
π
28
)
.
{\displaystyle \tan({\frac {\pi }{28}}),\qquad \tan({\frac {9\pi }{28}}),\qquad \tan({\frac {25\pi }{28}}).~}
x
3
−
(
4
+
7
)
x
2
−
(
11
+
4
7
)
x
+
8
+
3
7
{\displaystyle x^{3}-(4+{\sqrt {7}})x^{2}-(11+4{\sqrt {7}})x+8+3{\sqrt {7}}~}
Est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
5
π
28
)
,
tan
(
13
π
28
)
,
tan
(
17
π
28
)
.
{\displaystyle \tan({\frac {5\pi }{28}}),\qquad \tan({\frac {13\pi }{28}}),\qquad \tan({\frac {17\pi }{28}}).~}
x
3
+
(
4
+
7
)
x
2
−
(
11
+
4
7
)
x
−
8
−
3
7
{\displaystyle x^{3}+(4+{\sqrt {7}})x^{2}-(11+4{\sqrt {7}})x-8-3{\sqrt {7}}~}
Est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
11
π
28
)
,
tan
(
15
π
28
)
,
tan
(
23
π
28
)
.
{\displaystyle \tan({\frac {11\pi }{28}}),\qquad \tan({\frac {15\pi }{28}}),\qquad \tan({\frac {23\pi }{28}}).~}
x
3
+
(
4
−
7
)
x
2
−
(
11
−
4
7
)
x
−
8
+
3
7
{\displaystyle x^{3}+(4-{\sqrt {7}})x^{2}-(11-4{\sqrt {7}})x-8+3{\sqrt {7}}~}
Est le polynôme minimal dans
Q
(
7
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {7}})[X]}
tan
(
3
π
28
)
,
tan
(
19
π
28
)
,
tan
(
27
π
28
)
.
{\displaystyle \tan({\frac {3\pi }{28}}),\qquad \tan({\frac {19\pi }{28}}),\qquad \tan({\frac {27\pi }{28}}).~}
x
3
−
3
(
2
−
3
)
x
2
−
3
x
+
2
−
3
{\displaystyle x^{3}-3(2-{\sqrt {3}})x^{2}-3x+2-{\sqrt {3}}~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
π
36
)
,
tan
(
13
π
36
)
,
tan
(
25
π
36
)
.
{\displaystyle \tan({\frac {\pi }{36}}),\qquad \tan({\frac {13\pi }{36}}),\qquad \tan({\frac {25\pi }{36}}).~}
x
3
−
3
(
2
+
3
)
x
2
−
3
x
+
2
+
3
{\displaystyle x^{3}-3(2+{\sqrt {3}})x^{2}-3x+2+{\sqrt {3}}~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
5
π
36
)
,
tan
(
17
π
36
)
,
tan
(
29
π
36
)
.
{\displaystyle \tan({\frac {5\pi }{36}}),\qquad \tan({\frac {17\pi }{36}}),\qquad \tan({\frac {29\pi }{36}}).~}
x
3
+
3
(
2
+
3
)
x
2
−
3
x
−
2
−
3
{\displaystyle x^{3}+3(2+{\sqrt {3}})x^{2}-3x-2-{\sqrt {3}}~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
7
π
36
)
,
tan
(
19
π
36
)
,
tan
(
31
π
36
)
.
{\displaystyle \tan({\frac {7\pi }{36}}),\qquad \tan({\frac {19\pi }{36}}),\qquad \tan({\frac {31\pi }{36}}).~}
x
3
+
3
(
2
−
3
)
x
2
−
3
x
−
2
+
3
{\displaystyle x^{3}+3(2-{\sqrt {3}})x^{2}-3x-2+{\sqrt {3}}~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
11
π
36
)
,
tan
(
23
π
36
)
,
tan
(
35
π
36
)
.
{\displaystyle \tan({\frac {11\pi }{36}}),\qquad \tan({\frac {23\pi }{36}}),\qquad \tan({\frac {35\pi }{36}}).~}
x
4
−
10
x
2
+
5
{\displaystyle x^{4}-10x^{2}+5~}
Est le polynôme minimal des nombres :
tan
(
π
5
)
,
tan
(
2
π
5
)
,
tan
(
3
π
5
)
tan
(
4
π
5
)
.
{\displaystyle \tan({\frac {\pi }{5}}),\qquad \tan({\frac {2\pi }{5}}),\qquad \tan({\frac {3\pi }{5}})\qquad \tan({\frac {4\pi }{5}}).~}
x
4
−
2
x
2
+
2
{\displaystyle x^{4}-2x^{2}+2~}
Est le polynôme minimal des nombres :
tan
(
π
10
)
,
tan
(
3
π
10
)
,
tan
(
5
π
10
)
tan
(
7
π
10
)
.
{\displaystyle \tan({\frac {\pi }{10}}),\qquad \tan({\frac {3\pi }{10}}),\qquad \tan({\frac {5\pi }{10}})\qquad \tan({\frac {7\pi }{10}}).~}
x
4
−
4
x
3
−
14
x
2
−
4
x
+
1
{\displaystyle x^{4}-4x^{3}-14x^{2}-4x+1~}
Est le polynôme minimal des nombres :
tan
(
π
20
)
,
tan
(
9
π
20
)
,
tan
(
13
π
20
)
tan
(
17
π
20
)
.
{\displaystyle \tan({\frac {\pi }{20}}),\qquad \tan({\frac {9\pi }{20}}),\qquad \tan({\frac {13\pi }{20}})\qquad \tan({\frac {17\pi }{20}}).~}
x
4
+
4
x
3
−
14
x
2
+
4
x
+
1
{\displaystyle x^{4}+4x^{3}-14x^{2}+4x+1~}
Est le polynôme minimal des nombres :
tan
(
3
π
20
)
,
tan
(
7
π
20
)
,
tan
(
11
π
20
)
tan
(
19
π
20
)
.
{\displaystyle \tan({\frac {3\pi }{20}}),\qquad \tan({\frac {7\pi }{20}}),\qquad \tan({\frac {11\pi }{20}})\qquad \tan({\frac {19\pi }{20}}).~}
x
4
+
8
x
3
+
2
x
2
−
8
x
+
1
{\displaystyle x^{4}+8x^{3}+2x^{2}-8x+1~}
Est le polynôme minimal des nombres :
tan
(
π
24
)
,
tan
(
5
π
24
)
,
tan
(
13
π
24
)
tan
(
17
π
24
)
.
{\displaystyle \tan({\frac {\pi }{24}}),\qquad \tan({\frac {5\pi }{24}}),\qquad \tan({\frac {13\pi }{24}})\qquad \tan({\frac {17\pi }{24}}).~}
x
4
−
8
x
3
+
2
x
2
+
8
x
+
1
{\displaystyle x^{4}-8x^{3}+2x^{2}+8x+1~}
Est le polynôme minimal des nombres :
tan
(
7
π
24
)
,
tan
(
11
π
24
)
,
tan
(
19
π
24
)
tan
(
23
π
24
)
.
{\displaystyle \tan({\frac {7\pi }{24}}),\qquad \tan({\frac {11\pi }{24}}),\qquad \tan({\frac {19\pi }{24}})\qquad \tan({\frac {23\pi }{24}}).~}
x
4
+
4
x
3
2
+
2
x
2
−
4
x
2
+
1
{\displaystyle x^{4}+4x^{3}{\sqrt {2}}+2x^{2}-4x{\sqrt {2}}+1~}
Est le polynôme minimal dans
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
tan
(
π
16
)
,
tan
(
3
π
16
)
,
tan
(
9
π
16
)
tan
(
11
π
16
)
.
{\displaystyle \tan({\frac {\pi }{16}}),\qquad \tan({\frac {3\pi }{16}}),\qquad \tan({\frac {9\pi }{16}})\qquad \tan({\frac {11\pi }{16}}).~}
x
4
−
4
x
3
2
+
2
x
2
+
4
x
2
+
1
{\displaystyle x^{4}-4x^{3}{\sqrt {2}}+2x^{2}+4x{\sqrt {2}}+1~}
Est le polynôme minimal dans
Q
(
2
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {2}})[X]}
tan
(
5
π
16
)
,
tan
(
7
π
16
)
,
tan
(
13
π
16
)
tan
(
15
π
16
)
.
{\displaystyle \tan({\frac {5\pi }{16}}),\qquad \tan({\frac {7\pi }{16}}),\qquad \tan({\frac {13\pi }{16}})\qquad \tan({\frac {15\pi }{16}}).~}
x
4
+
6
x
3
3
+
8
x
2
−
2
x
3
−
1
{\displaystyle x^{4}+6x^{3}{\sqrt {3}}+8x^{2}-2x{\sqrt {3}}-1~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
2
π
15
)
,
tan
(
8
π
15
)
,
tan
(
11
π
15
)
tan
(
14
π
15
)
.
{\displaystyle \tan({\frac {2\pi }{15}}),\qquad \tan({\frac {8\pi }{15}}),\qquad \tan({\frac {11\pi }{15}})\qquad \tan({\frac {14\pi }{15}}).~}
x
4
−
6
x
3
3
+
8
x
2
+
2
x
3
−
1
{\displaystyle x^{4}-6x^{3}{\sqrt {3}}+8x^{2}+2x{\sqrt {3}}-1~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
π
15
)
,
tan
(
4
π
15
)
,
tan
(
7
π
15
)
tan
(
13
π
15
)
.
{\displaystyle \tan({\frac {\pi }{15}}),\qquad \tan({\frac {4\pi }{15}}),\qquad \tan({\frac {7\pi }{15}})\qquad \tan({\frac {13\pi }{15}}).~}
x
4
−
2
x
3
3
−
8
x
2
+
6
x
3
−
1
{\displaystyle x^{4}-2x^{3}{\sqrt {3}}-8x^{2}+6x{\sqrt {3}}-1~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
π
30
)
,
tan
(
7
π
30
)
,
tan
(
13
π
30
)
tan
(
19
π
30
)
.
{\displaystyle \tan({\frac {\pi }{30}}),\qquad \tan({\frac {7\pi }{30}}),\qquad \tan({\frac {13\pi }{30}})\qquad \tan({\frac {19\pi }{30}}).~}
x
4
+
2
x
3
3
−
8
x
2
−
6
x
3
−
1
{\displaystyle x^{4}+2x^{3}{\sqrt {3}}-8x^{2}-6x{\sqrt {3}}-1~}
Est le polynôme minimal dans
Q
(
3
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {3}})[X]}
tan
(
11
π
30
)
,
tan
(
17
π
30
)
,
tan
(
23
π
30
)
tan
(
29
π
30
)
.
{\displaystyle \tan({\frac {11\pi }{30}}),\qquad \tan({\frac {17\pi }{30}}),\qquad \tan({\frac {23\pi }{30}})\qquad \tan({\frac {29\pi }{30}}).~}
x
5
+
3
x
4
11
+
22
x
3
+
2
x
2
11
−
11
x
−
11
{\displaystyle x^{5}+3x^{4}{\sqrt {11}}+22x^{3}+2x^{2}{\sqrt {11}}-11x-{\sqrt {11}}~}
Est le polynôme minimal dans
Q
(
11
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {11}})[X]}
tan
(
2
π
11
)
,
tan
(
6
π
11
)
,
tan
(
7
π
11
)
tan
(
8
π
11
)
,
tan
(
10
π
11
)
.
{\displaystyle \tan({\frac {2\pi }{11}}),\qquad \tan({\frac {6\pi }{11}}),\qquad \tan({\frac {7\pi }{11}})\qquad \tan({\frac {8\pi }{11}}),\qquad \tan({\frac {10\pi }{11}}).~}
x
5
−
3
x
4
11
+
22
x
3
−
2
x
2
11
−
11
x
+
11
{\displaystyle x^{5}-3x^{4}{\sqrt {11}}+22x^{3}-2x^{2}{\sqrt {11}}-11x+{\sqrt {11}}~}
Est le polynôme minimal dans
Q
(
11
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {11}})[X]}
tan
(
π
11
)
,
tan
(
3
π
11
)
,
tan
(
4
π
11
)
tan
(
5
π
11
)
,
tan
(
9
π
11
)
.
{\displaystyle \tan({\frac {\pi }{11}}),\qquad \tan({\frac {3\pi }{11}}),\qquad \tan({\frac {4\pi }{11}})\qquad \tan({\frac {5\pi }{11}}),\qquad \tan({\frac {9\pi }{11}}).~}
x
5
−
x
4
11
−
2
x
3
+
2
x
2
11
−
3
x
+
1
11
{\displaystyle x^{5}-x^{4}{\sqrt {11}}-2x^{3}+2x^{2}{\sqrt {11}}-3x+{\frac {1}{\sqrt {11}}}~}
Est le polynôme minimal dans
Q
(
11
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {11}})[X]}
tan
(
π
22
)
,
tan
(
3
π
22
)
,
tan
(
5
π
22
)
tan
(
9
π
22
)
,
tan
(
15
π
22
)
.
{\displaystyle \tan({\frac {\pi }{22}}),\qquad \tan({\frac {3\pi }{22}}),\qquad \tan({\frac {5\pi }{22}})\qquad \tan({\frac {9\pi }{22}}),\qquad \tan({\frac {15\pi }{22}}).~}
x
5
+
x
4
11
−
2
x
3
−
2
x
2
11
−
3
x
−
1
11
{\displaystyle x^{5}+x^{4}{\sqrt {11}}-2x^{3}-2x^{2}{\sqrt {11}}-3x-{\frac {1}{\sqrt {11}}}~}
Est le polynôme minimal dans
Q
(
11
)
[
X
]
{\displaystyle \mathbb {Q} ({\sqrt {11}})[X]}
tan
(
7
π
22
)
,
tan
(
13
π
22
)
,
tan
(
17
π
22
)
tan
(
19
π
22
)
,
tan
(
21
π
22
)
.
{\displaystyle \tan({\frac {7\pi }{22}}),\qquad \tan({\frac {13\pi }{22}}),\qquad \tan({\frac {17\pi }{22}})\qquad \tan({\frac {19\pi }{22}}),\qquad \tan({\frac {21\pi }{22}}).~}
x
5
−
5
x
4
5
−
2
5
−
10
x
3
+
10
x
2
5
−
2
5
+
5
x
−
5
−
2
5
{\displaystyle x^{5}-5x^{4}{\sqrt {5-2{\sqrt {5}}}}-10x^{3}+10x^{2}{\sqrt {5-2{\sqrt {5}}}}+5x-{\sqrt {5-2{\sqrt {5}}}}~}
Est le polynôme minimal dans
Q
(
5
,
5
−
2
5
,
5
+
2
5
,
25
−
10
5
,
25
+
10
5
)
[
X
]
{\displaystyle \mathbb {Q} \left({\sqrt {5}},{\sqrt {5-2{\sqrt {5}}}},{\sqrt {5+2{\sqrt {5}}}},{\sqrt {25-10{\sqrt {5}}}},{\sqrt {25+10{\sqrt {5}}}}\right)[X]}
tan
(
π
25
)
,
tan
(
6
π
25
)
,
tan
(
11
π
25
)
tan
(
16
π
25
)
,
tan
(
21
π
25
)
.
{\displaystyle \tan({\frac {\pi }{25}}),\qquad \tan({\frac {6\pi }{25}}),\qquad \tan({\frac {11\pi }{25}})\qquad \tan({\frac {16\pi }{25}}),\qquad \tan({\frac {21\pi }{25}}).~}
x
5
−
5
x
4
5
+
2
5
−
10
x
3
+
10
x
2
5
+
2
5
+
5
x
−
5
+
2
5
{\displaystyle x^{5}-5x^{4}{\sqrt {5+2{\sqrt {5}}}}-10x^{3}+10x^{2}{\sqrt {5+2{\sqrt {5}}}}+5x-{\sqrt {5+2{\sqrt {5}}}}~}
Est le polynôme minimal dans
Q
(
5
,
5
−
2
5
,
5
+
2
5
,
25
−
10
5
,
25
+
10
5
)
[
X
]
{\displaystyle \mathbb {Q} \left({\sqrt {5}},{\sqrt {5-2{\sqrt {5}}}},{\sqrt {5+2{\sqrt {5}}}},{\sqrt {25-10{\sqrt {5}}}},{\sqrt {25+10{\sqrt {5}}}}\right)[X]}
tan
(
2
π
25
)
,
tan
(
7
π
25
)
,
tan
(
12
π
25
)
tan
(
17
π
25
)
,
tan
(
22
π
25
)
.
{\displaystyle \tan({\frac {2\pi }{25}}),\qquad \tan({\frac {7\pi }{25}}),\qquad \tan({\frac {12\pi }{25}})\qquad \tan({\frac {17\pi }{25}}),\qquad \tan({\frac {22\pi }{25}}).~}
x
5
+
5
x
4
5
+
2
5
−
10
x
3
−
10
x
2
5
+
2
5
+
5
x
+
5
+
2
5
{\displaystyle x^{5}+5x^{4}{\sqrt {5+2{\sqrt {5}}}}-10x^{3}-10x^{2}{\sqrt {5+2{\sqrt {5}}}}+5x+{\sqrt {5+2{\sqrt {5}}}}~}
Est le polynôme minimal dans
Q
(
5
,
5
−
2
5
,
5
+
2
5
,
25
−
10
5
,
25
+
10
5
)
[
X
]
{\displaystyle \mathbb {Q} \left({\sqrt {5}},{\sqrt {5-2{\sqrt {5}}}},{\sqrt {5+2{\sqrt {5}}}},{\sqrt {25-10{\sqrt {5}}}},{\sqrt {25+10{\sqrt {5}}}}\right)[X]}
tan
(
3
π
25
)
,
tan
(
8
π
25
)
,
tan
(
13
π
25
)
tan
(
18
π
25
)
,
tan
(
23
π
25
)
.
{\displaystyle \tan({\frac {3\pi }{25}}),\qquad \tan({\frac {8\pi }{25}}),\qquad \tan({\frac {13\pi }{25}})\qquad \tan({\frac {18\pi }{25}}),\qquad \tan({\frac {23\pi }{25}}).~}
x
5
+
5
x
4
5
−
2
5
−
10
x
3
−
10
x
2
5
−
2
5
+
5
x
+
5
−
2
5
{\displaystyle x^{5}+5x^{4}{\sqrt {5-2{\sqrt {5}}}}-10x^{3}-10x^{2}{\sqrt {5-2{\sqrt {5}}}}+5x+{\sqrt {5-2{\sqrt {5}}}}~}
Est le polynôme minimal dans
Q
(
5
,
5
−
2
5
,
5
+
2
5
,
25
−
10
5
,
25
+
10
5
)
[
X
]
{\displaystyle \mathbb {Q} \left({\sqrt {5}},{\sqrt {5-2{\sqrt {5}}}},{\sqrt {5+2{\sqrt {5}}}},{\sqrt {25-10{\sqrt {5}}}},{\sqrt {25+10{\sqrt {5}}}}\right)[X]}
tan
(
4
π
25
)
,
tan
(
9
π
25
)
,
tan
(
14
π
25
)
tan
(
19
π
25
)
,
tan
(
24
π
25
)
.
{\displaystyle \tan({\frac {4\pi }{25}}),\qquad \tan({\frac {9\pi }{25}}),\qquad \tan({\frac {14\pi }{25}})\qquad \tan({\frac {19\pi }{25}}),\qquad \tan({\frac {24\pi }{25}}).~}
Etc...
On trouvera ci-dessous quelques formules relatives aux relations entre coefficients et racines d'un polynome annulateur des nombres trigonométriques traités dans cet article. Certaines de ces formules font apparaître au second membre la racine d'un diviseur du dénominateur se trouvant dans l'argument des fonctions trigonométriques au premier membre. Ce qui permet d'entrevoir les raisons pour lesquelles, on trouve dans les coefficients du polynôme minimal de telle racines.
∀
n
∈
N
∗
∑
k
=
1
n
cos
(
2
k
π
2
n
+
1
)
=
−
1
2
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \sum _{k=1}^{n}\cos \left({\frac {2k\pi }{2n+1}}\right)=-{\frac {1}{2}}~}
∀
n
∈
N
∗
∑
k
=
1
n
cos
(
(
2
k
−
1
)
π
2
n
)
=
0
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \sum _{k=1}^{n}\cos \left({\frac {(2k-1)\pi }{2n}}\right)=0~}
(Dans ce qui suit Ent(x) désigne la partie entière de x)
∀
n
∈
N
∗
∏
k
=
1
n
cos
(
k
π
2
n
+
1
)
=
1
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\cos \left({\frac {k\pi }{2n+1}}\right)={\frac {1}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
sin
(
k
π
2
n
+
1
)
=
2
n
+
1
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\sin \left({\frac {k\pi }{2n+1}}\right)={\frac {\sqrt {2n+1}}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
tan
(
k
π
2
n
+
1
)
=
2
n
+
1
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\tan \left({\frac {k\pi }{2n+1}}\right)={\sqrt {2n+1}}~}
∀
n
∈
N
∗
∏
k
=
1
n
cos
(
2
k
π
2
n
+
1
)
=
(
−
1
)
E
n
t
(
n
−
1
2
)
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\cos \left({\frac {2k\pi }{2n+1}}\right)={\frac {(-1)^{Ent({\frac {n-1}{2}})}}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
sin
(
2
k
π
2
n
+
1
)
=
2
n
+
1
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\sin \left({\frac {2k\pi }{2n+1}}\right)={\frac {\sqrt {2n+1}}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
tan
(
2
k
π
2
n
+
1
)
=
(
−
1
)
E
n
t
(
n
−
1
2
)
2
n
+
1
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\tan \left({\frac {2k\pi }{2n+1}}\right)=(-1)^{Ent({\frac {n-1}{2}})}{\sqrt {2n+1}}~}
∀
n
∈
N
−
{
0
,
1
}
∏
k
=
1
n
−
1
cos
(
k
π
2
n
)
=
n
2
n
−
1
{\displaystyle \forall n\in \mathbb {N} -\left\{0,1\right\}\qquad \prod _{k=1}^{n-1}\cos \left({\frac {k\pi }{2n}}\right)={\frac {\sqrt {n}}{2^{n-1}}}~}
∀
n
∈
N
−
{
0
,
1
}
∏
k
=
1
n
−
1
sin
(
k
π
2
n
)
=
n
2
n
−
1
{\displaystyle \forall n\in \mathbb {N} -\left\{0,1\right\}\qquad \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{2n}}\right)={\frac {\sqrt {n}}{2^{n-1}}}~}
∀
n
∈
N
−
{
0
,
1
}
∏
k
=
1
n
−
1
tan
(
k
π
2
n
)
=
1
{\displaystyle \forall n\in \mathbb {N} -\left\{0,1\right\}\qquad \prod _{k=1}^{n-1}\tan \left({\frac {k\pi }{2n}}\right)=1~}
∀
n
∈
N
∗
∏
k
=
1
2
n
cos
(
k
π
2
n
+
1
)
=
1
(
−
4
)
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{2n}\cos \left({\frac {k\pi }{2n+1}}\right)={\frac {1}{(-4)^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
2
n
sin
(
k
π
2
n
+
1
)
=
n
4
n
−
1
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{2n}\sin \left({\frac {k\pi }{2n+1}}\right)={\frac {n}{4^{n-1}}}~}
∀
n
∈
N
∗
∏
k
=
1
2
n
tan
(
k
π
2
n
+
1
)
=
(
−
1
)
n
(
2
n
+
1
)
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{2n}\tan \left({\frac {k\pi }{2n+1}}\right)=(-1)^{n}(2n+1)~}
∀
n
∈
N
∗
∏
k
=
1
n
cos
(
(
2
k
−
1
)
π
4
n
)
=
2
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\cos \left({\frac {(2k-1)\pi }{4n}}\right)={\frac {\sqrt {2}}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
sin
(
(
2
k
−
1
)
π
4
n
)
=
2
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\sin \left({\frac {(2k-1)\pi }{4n}}\right)={\frac {\sqrt {2}}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
tan
(
(
2
k
−
1
)
π
4
n
)
=
1
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\tan \left({\frac {(2k-1)\pi }{4n}}\right)=1~}
∀
n
∈
N
∗
∏
k
=
1
n
cos
(
(
2
k
−
1
)
π
4
n
+
2
)
=
2
n
+
1
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\cos \left({\frac {(2k-1)\pi }{4n+2}}\right)={\frac {\sqrt {2n+1}}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
sin
(
(
2
k
−
1
)
π
4
n
+
2
)
=
1
2
n
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\sin \left({\frac {(2k-1)\pi }{4n+2}}\right)={\frac {1}{2^{n}}}~}
∀
n
∈
N
∗
∏
k
=
1
n
tan
(
(
2
k
−
1
)
π
4
n
+
2
)
=
1
2
n
+
1
{\displaystyle \forall n\in \mathbb {N} ^{*}\qquad \prod _{k=1}^{n}\tan \left({\frac {(2k-1)\pi }{4n+2}}\right)={\frac {1}{\sqrt {2n+1}}}~}
Dans la suite, on désignera par P(n), l'ensemble des nombres entiers inférieurs à n et premier avec n. Par exemple :
P
(
15
)
=
{
1
,
2
,
4
,
7
,
8
,
11
,
13
,
14
}
{\displaystyle P(15)=\left\{1,2,4,7,8,11,13,14\right\}}
On désignera aussi par
P
{\displaystyle \mathbb {P} }
∀
n
∈
N
−
{
0
,
1
}
∏
k
∈
P
(
4
n
)
tan
(
k
π
4
n
)
=
1
{\displaystyle \forall n\in \mathbb {N} -\left\{0,1\right\}\qquad \prod _{k\in P(4n)}\tan \left({\frac {k\pi }{4n}}\right)=1~}
∀
p
∈
P
−
{
2
}
,
∀
n
∈
N
∗
∏
k
∈
P
(
p
n
)
tan
(
k
π
p
n
)
=
(
−
1
)
p
−
1
2
p
{\displaystyle \forall p\in \mathbb {P} -\left\{2\right\},\forall n\in \mathbb {N} ^{*}\qquad \prod _{k\in P(p^{n})}\tan \left({\frac {k\pi }{p^{n}}}\right)=(-1)^{\frac {p-1}{2}}p~}
∀
p
∈
P
−
{
2
}
,
∀
n
∈
N
∗
∏
k
∈
P
(
2
p
n
)
tan
(
k
π
2
p
n
)
=
(
−
1
)
p
−
1
2
p
{\displaystyle \forall p\in \mathbb {P} -\left\{2\right\},\forall n\in \mathbb {N} ^{*}\qquad \prod _{k\in P(2p^{n})}\tan \left({\frac {k\pi }{2p^{n}}}\right)={\frac {(-1)^{\frac {p-1}{2}}}{p}}~}
∏
k
∈
P
(
n
)
tan
(
k
π
n
)
=
1
{\displaystyle \prod _{k\in P(n)}\tan \left({\frac {k\pi }{n}}\right)=1~}
∏
k
∈
P
(
4
n
)
2
cos
(
k
π
4
n
)
=
1
{\displaystyle \prod _{k\in P(4n)}2\cos \left({\frac {k\pi }{4n}}\right)=1~}
∀
n
∈
N
−
{
0
,
1
,
2
}
∏
k
∈
P
(
2
n
)
2
cos
(
k
π
2
n
)
=
2
{\displaystyle \forall n\in \mathbb {N} -\left\{0,1,2\right\}\qquad \prod _{k\in P(2^{n})}2\cos \left({\frac {k\pi }{2^{n}}}\right)=2~}
∀
p
∈
P
−
{
2
}
,
∀
n
∈
N
∗
∏
k
∈
P
(
p
n
)
2
cos
(
k
π
p
n
)
=
(
−
1
)
p
−
1
2
{\displaystyle \forall p\in \mathbb {P} -\left\{2\right\},\forall n\in \mathbb {N} ^{*}\qquad \prod _{k\in P(p^{n})}2\cos \left({\frac {k\pi }{p^{n}}}\right)=(-1)^{\frac {p-1}{2}}~}
∀
p
∈
P
−
{
2
}
,
∀
n
∈
N
∗
∏
k
∈
P
(
2
p
n
)
2
cos
(
k
π
2
p
n
)
=
(
−
1
)
p
−
1
2
p
{\displaystyle \forall p\in \mathbb {P} -\left\{2\right\},\forall n\in \mathbb {N} ^{*}\qquad \prod _{k\in P(2p^{n})}2\cos \left({\frac {k\pi }{2p^{n}}}\right)=(-1)^{\frac {p-1}{2}}p~}
∏
k
∈
P
(
n
)
2
cos
(
k
π
n
)
=
1
{\displaystyle \prod _{k\in P(n)}2\cos \left({\frac {k\pi }{n}}\right)=1~}
∏
k
≤
n
,
k
∈
P
(
2
n
+
1
)
tan
(
k
π
2
n
+
1
)
=
p
{\displaystyle \prod _{k\leq n,k\in P(2n+1)}\tan \left({\frac {k\pi }{2n+1}}\right)={\sqrt {p}}~}
∏
k
≤
n
,
k
∈
P
(
2
n
+
1
)
tan
(
k
π
2
n
+
1
)
=
1
{\displaystyle \prod _{k\leq n,k\in P(2n+1)}\tan \left({\frac {k\pi }{2n+1}}\right)=1~}
Etc...
{{Portail|mathématiques}}
[[Catégorie:Trigonométrie]]
[[Catégorie:Polynôme remarquable|minimal]]
[[Catégorie:Liste en rapport avec les mathématiques]]
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