Dieser Artikel ist im Entstehen und noch nicht Bestandteil der freien Enzyklopädie Wikipedia .
Solltest du über eine Suchmaschine darauf gestoßen sein, bedenke, dass der Text noch unvollständig sein und Fehler oder ungeprüfte Aussagen enthalten kann. Wenn du Fragen zum Thema hast, nimm Kontakt mit dem Autor Mathemix auf.
Dieser Benutzer ist seit 1978 Tagen bei Wikipedia
Unterseiten:
/Algebra
/Analysis
/Differentialgeometrie
/Physik
https://de.wikipedia.org/wiki/Kategorie:Wikipedia:Formatvorlage
https://de.wikipedia.org/wiki/Kategorie:Vorlage:Infobox
https://de.wikipedia.org/wiki/Wikipedia:Starthilfe
Quellen:
van der Waerden, Algebra I, Springer-Verlag
3
x
1
+
2
x
2
−
x
3
=
1
2
x
1
−
2
x
2
+
4
x
3
=
−
2
−
x
1
+
1
2
x
2
−
x
3
=
0
{\displaystyle {\begin{matrix}3x_{1}&+&2x_{2}&-&x_{3}&=&1\\2x_{1}&-&2x_{2}&+&4x_{3}&=&-2\\-x_{1}&+&{1 \over 2}x_{2}&-&x_{3}&=&0\end{matrix}}}
a
11
x
1
+
a
12
x
2
+
⋯
+
a
1
n
x
n
=
b
1
a
21
x
1
+
a
22
x
2
+
⋯
+
a
2
n
x
n
=
b
2
⋮
a
m
1
x
1
+
a
m
2
x
2
+
⋯
+
a
m
n
x
n
=
b
m
{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}\,+&\cdots &+\,a_{1n}x_{n}&=&b_{1}\\a_{21}x_{1}+a_{22}x_{2}\,+&\cdots &+\,a_{2n}x_{n}&=&b_{2}\\&&&\vdots &\\a_{m1}x_{1}+a_{m2}x_{2}\,+&\cdots &+\,a_{mn}x_{n}&=&b_{m}\\\end{matrix}}}
x
1
{\displaystyle x_{1}}
p1
(
x
2
−
5
)
2
−
24
{\displaystyle \left(x^{2}-5\right)^{2}-24}
Standardskalarprodukt im Rn n 3
Bearbeiten
(erscheint richtig in der Überschrift)
2D:
∫
0
R
∫
0
2
π
r
d
ϕ
1
d
r
=
π
R
2
{\displaystyle \int _{0}^{R}\int _{0}^{2\pi }r\mathrm {d} \phi _{1}\mathrm {d} r=\pi R^{2}}
3D:
∫
0
R
∫
0
2
π
∫
0
π
r
2
sin
(
ϕ
1
)
d
ϕ
1
d
ϕ
2
d
r
=
4
π
R
3
3
{\displaystyle \int _{0}^{R}\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin(\phi _{1}){\text{d}}\phi _{1}{\text{d}}\phi _{2}{\text{d}}r={\frac {4\pi R^{3}}{3}}}
4D:
∫
0
R
∫
0
2
π
∫
0
π
∫
0
π
r
3
sin
2
(
ϕ
1
)
sin
(
ϕ
2
)
d
ϕ
1
d
ϕ
2
d
ϕ
3
d
r
=
π
2
R
4
2
{\displaystyle \int _{0}^{R}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{\pi }r^{3}\sin ^{2}(\phi _{1})\sin(\phi _{2}){\text{d}}\phi _{1}{\text{d}}\phi _{2}{\text{d}}\phi _{3}{\text{d}}r={\frac {\pi ^{2}R^{4}}{2}}}
Vaughan F. R. Jones: A polynomial invariant for knots via von Neumann algebras . In: Hyman Bass, Meyer Jerison, Calvin C. Moore (Hrsg.): Bulletin of the American Mathematical Society (New Series) . Vol. 12, Nr. 1 . American Mathematical Society, 1985, ISSN 0273-0979 S. 103–111 , doi :10.1090/S0273-0979-1985-15304-2 ams.org [PDF; abgerufen am 2. Dezember 2012]).
Q
{\displaystyle \textstyle \mathbb {Q} }
Q
(
2
)
=
{
a
+
b
2
∣
a
,
b
∈
Q
}
{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})=\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \}}
{
1
,
2
{\displaystyle \textstyle 1,{\sqrt {2}}}
Q
(
2
,
3
)
=
{
a
+
b
2
+
c
3
+
d
6
∣
a
,
b
,
c
,
d
∈
Q
}
{\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})=\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \}}
{
1
,
2
,
3
,
6
{\displaystyle \textstyle 1,{\sqrt {2}},{\sqrt {3}},{\sqrt {6}}}
5
±
2
6
=
(
2
±
3
)
2
{\displaystyle 5\pm 2{\sqrt {6}}=({\sqrt {2}}\pm {\sqrt {3}})^{2}}
x
1
=
2
+
3
{\displaystyle x_{1}={\sqrt {2}}+{\sqrt {3}}}
2
=
1
2
(
x
1
3
−
9
x
1
)
{\displaystyle {\sqrt {2}}={\tfrac {1}{2}}(x_{1}^{3}-9x_{1})}
3
=
−
1
2
(
x
1
3
−
11
x
1
)
{\displaystyle {\sqrt {3}}=-{\tfrac {1}{2}}(x_{1}^{3}-11x_{1})}
1
a
+
b
2
=
(
a
−
b
2
)
(
a
+
b
2
)
⋅
(
a
−
b
2
)
=
(
a
−
b
2
)
(
a
2
−
2
b
2
)
=
a
(
a
2
−
2
b
2
)
+
−
b
(
a
2
−
2
b
2
)
2
{\displaystyle {\frac {1}{a+b{\sqrt {2}}}}={\frac {(a-b{\sqrt {2}})}{(a+b{\sqrt {2}})\cdot (a-b{\sqrt {2}})}}={\frac {(a-b{\sqrt {2}})}{(a^{2}-2b^{2})}}={\frac {a}{(a^{2}-2b^{2})}}+{\frac {-b}{(a^{2}-2b^{2})}}{\sqrt {2}}}
'kursiv' (???)
Beispiel (???)
fett
(
x
1
,
x
2
,
x
3
,
x
4
)
↦
.
.
.
{\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)\mapsto ...}
Nr.
Permutation
Nr.
Permutation
Nr.
Permutation
Nr.
Permutation
1
(
x
1
,
x
2
,
x
3
,
x
4
)
{\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)}
7
(
x
2
,
x
1
,
x
3
,
x
4
)
{\displaystyle \left(x_{2},x_{1},x_{3},x_{4}\right)}
13
(
x
3
,
x
1
,
x
2
,
x
4
)
{\displaystyle \left(x_{3},x_{1},x_{2},x_{4}\right)}
19
(
x
4
,
x
1
,
x
2
,
x
3
)
{\displaystyle \left(x_{4},x_{1},x_{2},x_{3}\right)}
2
(
x
1
,
x
2
,
x
4
,
x
3
)
{\displaystyle \left(x_{1},x_{2},x_{4},x_{3}\right)}
8
(
x
2
,
x
1
,
x
4
,
x
3
)
{\displaystyle \left(x_{2},x_{1},x_{4},x_{3}\right)}
14
(
x
3
,
x
1
,
x
4
,
x
2
)
{\displaystyle \left(x_{3},x_{1},x_{4},x_{2}\right)}
20
(
x
4
,
x
1
,
x
3
,
x
2
)
{\displaystyle \left(x_{4},x_{1},x_{3},x_{2}\right)}
3
(
x
1
,
x
3
,
x
2
,
x
4
)
{\displaystyle \left(x_{1},x_{3},x_{2},x_{4}\right)}
9
(
x
2
,
x
3
,
x
1
,
x
4
)
{\displaystyle \left(x_{2},x_{3},x_{1},x_{4}\right)}
15
(
x
3
,
x
2
,
x
1
,
x
4
)
{\displaystyle \left(x_{3},x_{2},x_{1},x_{4}\right)}
21
(
x
4
,
x
2
,
x
1
,
x
3
)
{\displaystyle \left(x_{4},x_{2},x_{1},x_{3}\right)}
4
(
x
1
,
x
3
,
x
4
,
x
2
)
{\displaystyle \left(x_{1},x_{3},x_{4},x_{2}\right)}
10
(
x
2
,
x
3
,
x
4
,
x
1
)
{\displaystyle \left(x_{2},x_{3},x_{4},x_{1}\right)}
16
(
x
3
,
x
2
,
x
4
,
x
1
)
{\displaystyle \left(x_{3},x_{2},x_{4},x_{1}\right)}
22
(
x
4
,
x
2
,
x
3
,
x
1
)
{\displaystyle \left(x_{4},x_{2},x_{3},x_{1}\right)}
5
(
x
1
,
x
4
,
x
2
,
x
3
)
{\displaystyle \left(x_{1},x_{4},x_{2},x_{3}\right)}
11
(
x
2
,
x
4
,
x
1
,
x
3
)
{\displaystyle \left(x_{2},x_{4},x_{1},x_{3}\right)}
17
(
x
3
,
x
4
,
x
1
,
x
2
)
{\displaystyle \left(x_{3},x_{4},x_{1},x_{2}\right)}
23
(
x
4
,
x
3
,
x
1
,
x
2
)
{\displaystyle \left(x_{4},x_{3},x_{1},x_{2}\right)}
6
(
x
1
,
x
4
,
x
3
,
x
2
)
{\displaystyle \left(x_{1},x_{4},x_{3},x_{2}\right)}
12
(
x
2
,
x
4
,
x
3
,
x
1
)
{\displaystyle \left(x_{2},x_{4},x_{3},x_{1}\right)}
18
(
x
3
,
x
4
,
x
2
,
x
1
)
{\displaystyle \left(x_{3},x_{4},x_{2},x_{1}\right)}
24
(
x
4
,
x
3
,
x
2
,
x
1
)
{\displaystyle \left(x_{4},x_{3},x_{2},x_{1}\right)}
Ohne Textstyle:
{
1
,
2
{\displaystyle 1,{\sqrt {2}}}
1
,
2
{\displaystyle \textstyle 1,{\sqrt {2}}}
n
→
i
⋅
x
→
=
b
i
{\displaystyle {\vec {n}}_{i}\cdot {\vec {x}}=b_{i}}
n
→
i
=
(
a
i
1
a
i
2
⋮
a
i
n
)
{\displaystyle {\vec {n}}_{i}={\begin{pmatrix}a_{i1}\\a_{i2}\\\vdots \\a_{in}\end{pmatrix}}\qquad }
(
2
−
1
)
⋅
(
x
1
x
2
)
=
4
{\displaystyle {\begin{pmatrix}2\\-1\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}=4}
Fehler beim Parsen (SVG (MathML kann über ein Browser-Plugin aktiviert werden): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „http://localhost:6011/de.wikipedia.org/v1/“:): {\displaystyle \begin{pmatrix} 1 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = == Zwischenräume == :<math>\sigma_1:\left(x_1,x_2,x_3,x_4\right)\mapsto\left(x_1,x_2,x_3,x_4\right)}
σ
2
:
(
x
1
,
x
2
,
x
3
,
x
4
)
↦
(
x
2
,
x
1
,
x
4
,
x
3
)
{\displaystyle \sigma _{2}:\left(x_{1},x_{2},x_{3},x_{4}\right)\mapsto \left(x_{2},x_{1},x_{4},x_{3}\right)}
Q
(
2
,
3
)
{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})}
p
1
(
x
1
)
=
x
1
,
p
2
(
x
1
)
=
x
2
,
p
3
(
x
1
)
=
x
3
,
p
4
(
x
1
)
=
x
4
,
⟹
σ
1
:
(
x
1
,
x
2
,
x
3
,
x
4
)
↦
(
x
1
,
x
2
,
x
3
,
x
4
)
{\displaystyle p_{1}(x_{1})=x_{1},\quad p_{2}(x_{1})=x_{2},\quad p_{3}(x_{1})=x_{3},\quad p_{4}(x_{1})=x_{4},\quad \Longrightarrow \quad \sigma _{1}:\left(x_{1},x_{2},x_{3},x_{4}\right)\mapsto \left(x_{1},x_{2},x_{3},x_{4}\right)}
griechische Buchstaben
Bearbeiten
p
1
(
x
1
)
=
x
1
,
p
2
(
x
1
)
=
x
2
,
p
3
(
x
1
)
=
x
3
,
p
4
(
x
1
)
=
x
4
,
⟹
σ
1
:
(
x
1
,
x
2
,
x
3
,
x
4
)
↦
(
x
1
,
x
2
,
x
3
,
x
4
)
{\displaystyle p_{1}(x_{1})=x_{1},\quad p_{2}(x_{1})=x_{2},\quad p_{3}(x_{1})=x_{3},\quad p_{4}(x_{1})=x_{4},\quad \Longrightarrow \quad \sigma _{1}:\left(x_{1},x_{2},x_{3},x_{4}\right)\mapsto \left(x_{1},x_{2},x_{3},x_{4}\right)}
Beispiel zum Satz vom primitiven Element
Galois-Resolvente
Nachweis der Stetigkeit der Funktion
f
(
x
)
=
2
x
+
3
{\displaystyle f(x)=2x+3}
an der Stelle
x
0
{\displaystyle x_{0}}
Herleitung
Mithilfe der Formel von Moivre-Binet lässt sich eine einfach Herleitung angeben. Denn für die Zahlen
Φ
,
Ψ
{\displaystyle \Phi ,\Psi }
n
>
0
{\displaystyle n>0}
−
Φ
<
0
<
−
Ψ
|
⋅
Ψ
2
n
>
0
{\displaystyle -\Phi <0<-\Psi \qquad |\cdot \Psi ^{2n}>0}
−
Φ
Ψ
2
n
<
−
Ψ
2
n
+
1
|
+
Φ
2
n
+
1
{\displaystyle -\Phi \Psi ^{2n}<-\Psi ^{2n+1}\qquad |+\Phi ^{2n+1}}
Φ
2
n
+
1
−
Φ
Ψ
2
n
=
Φ
(
Φ
2
n
−
Ψ
2
n
)
<
Φ
2
n
+
1
−
Ψ
2
n
+
1
|
:
(
Φ
2
n
−
Ψ
2
n
)
>
0
{\displaystyle \Phi ^{2n+1}-\Phi \Psi ^{2n}=\Phi (\Phi ^{2n}-\Psi ^{2n})<\Phi ^{2n+1}-\Psi ^{2n+1}\qquad |:(\Phi ^{2n}-\Psi ^{2n})>0\quad }
(1)
Φ
<
Φ
2
n
+
1
−
Ψ
2
n
+
1
Φ
2
n
−
Ψ
2
n
=
f
2
n
+
1
f
2
n
{\displaystyle \Phi <{\frac {\Phi ^{2n+1}-\Psi ^{2n+1}}{\Phi ^{2n}-\Psi ^{2n}}}={\frac {f_{2n+1}}{f_{2n}}}}
Φ
−
Ψ
{\displaystyle \Phi -\Psi }
−
Φ
<
0
<
−
Ψ
|
⋅
Ψ
2
n
−
1
<
0
{\displaystyle -\Phi <0<-\Psi \qquad |\cdot \Psi ^{2n-1}<0}
−
Φ
Ψ
2
n
−
1
>
−
Ψ
2
n
|
+
Φ
2
n
{\displaystyle -\Phi \Psi ^{2n-1}>-\Psi ^{2n}\qquad |+\Phi ^{2n}}
Φ
2
n
−
Φ
Ψ
2
n
−
1
=
Φ
(
Φ
2
n
−
1
−
Ψ
2
n
−
1
)
>
Φ
2
n
−
Ψ
2
n
|
:
(
Φ
2
n
−
1
−
Ψ
2
n
−
1
)
>
0
{\displaystyle \Phi ^{2n}-\Phi \Psi ^{2n-1}=\Phi (\Phi ^{2n-1}-\Psi ^{2n-1})>\Phi ^{2n}-\Psi ^{2n}\qquad |:(\Phi ^{2n-1}-\Psi ^{2n-1})>0}
Φ
>
Φ
2
n
−
Ψ
2
n
Φ
2
n
−
1
−
Ψ
2
n
−
1
=
f
2
n
f
2
n
−
1
{\displaystyle \Phi >{\frac {\Phi ^{2n}-\Psi ^{2n}}{\Phi ^{2n-1}-\Psi ^{2n-1}}}={\frac {f_{2n}}{f_{2n-1}}}\quad }
(2)
Die Ungleichungen (1) und (2) ergeben zusammen die Behauptung.
(
x
1
,
x
2
,
x
3
,
x
4
)
↦
(
x
1
,
x
2
,
x
3
,
x
4
)
{\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)\mapsto \left(x_{1},x_{2},x_{3},x_{4}\right)}
p
1
(
x
1
)
=
x
1
,
p
2
(
x
1
)
=
x
2
,
p
3
(
x
1
)
=
x
3
,
p
4
(
x
1
)
=
x
4
,
⟹
σ
1
:
(
x
1
,
x
2
,
x
3
,
x
4
)
↦
(
x
1
,
x
2
,
x
3
,
x
4
)
{\displaystyle p_{1}(x_{1})=x_{1},\quad p_{2}(x_{1})=x_{2},\quad p_{3}(x_{1})=x_{3},\quad p_{4}(x_{1})=x_{4},\quad \Longrightarrow \quad \sigma _{1}:\left(x_{1},x_{2},x_{3},x_{4}\right)\mapsto \left(x_{1},x_{2},x_{3},x_{4}\right)}
id
{\displaystyle \operatorname {id} }
[1]
https://de.wikiversity.org/wiki/Kurs:K%C3%B6rper-_und_Galoistheorie_(Osnabr%C3%BCck_2018-2019)/Vorlesung_17
↑ Nieper-Wißkirchen, Universität Augsburg: "Galoissche Theorie", S. 126, Proposition 4.8 und Beispiel, [1]
