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Aufgabensammlung zum Staatsexam in Algebra

Diese Seite enthät alle Staatsexamensaufgaben im Bereich Algebra für die Jahrgänge 2012 bis 2019 mit Lösungsvorschlägen, ab Herbst 2018 auch mit Kommentaren und Hinweisen versehen. Weiter unten finden Sie die Aufgaben nach Einzelthemen sortiert. Außerdem habe ich versucht, die Aufgaben nach dem Schwierigkeitsgrad einzuteilen (grün = standard, sollte keine bei Kenntnis des zu Grunde liegenden Stoffs keine Probleme bereiten, gelb = erfordert eine sichere Beherrschung des Stoffs und eventuell auch etwas Kreativität, rot = für Experten). Natürlich ist diese Einschätzung hochgradig subjektiv.

Eine alte Fassung der Seite finden Sie hier, sie enthält noch einige Lösungen von Aufgaben aus älteren Jahrgängen (vor 2012). Bitte beachten Sie, dass diese Seite von nun an nicht mehr aktualisiert wird.

I. Aufgaben in der Originalfassung

Jahrgänge 1972-2003 (zusammengestellt von Prof. W.-D. Geyer, Universität Erlangen-Nürnberg)
Jahrgänge 2000-2002 (PDF)
Jahrgänge 2003-2005 (PDF)
Jahrgänge 2006-2008 (PDF)
Jahrgänge 2009-2011 (PDF)
Jahrgänge 2012-2014 (PDF)
Jahrgänge 2015-2017 (PDF)
Jahrgänge 2018 und 2019 (PDF)

II. Lösungen

Jahrgang 2012

F12T1A1	F12T1A2	F12T1A3	F12T1A4	F12T1A5
F12T2A1	F12T2A2	F12T2A3	F12T2A4
F12T3A1	F12T3A2	F12T3A3	F12T3A4	F12T3A5
H12T1A1	H12T1A2	H12T1A3	H12T1A4	H12T1A5
H12T2A1	H12T2A2	H12T2A3	H12T2A4	H12T2A5
H12T3A1	H12T3A2	H12T3A3	H12T3A4	H12T3A5

Jahrgang 2013

F13T1A1	F13T1A2	F13T1A3	F13T1A4	F13T1A5
F13T2A1	F13T2A2	F13T2A3	F13T2A4	F13T2A5
F13T3A1	F13T3A2	F13T3A3	F13T3A4	F13T3A5
H13T1A1	H13T1A2	H13T1A3	H13T1A4	H13T1A5
H13T2A1	H13T2A2	H13T2A3	H13T2A4	H13T2A5
H13T3A1	H13T3A2	H13T3A3	H13T3A4	H13T3A5

Jahrgang 2014

F14T1A1	F14T1A2	F14T1A3	F14T1A4	F14T1A5
F14T2A1	F14T2A2	F14T2A3	F14T2A4	F14T2A5
F14T3A1	F14T3A2	F14T3A3	F14T3A4
H14T1A1	H14T1A2	H14T1A3	H14T1A4	H14T1A5
H14T2A1	H14T2A2	H14T2A3	H14T2A4	H14T2A5
H14T3A1	H14T3A2	H14T3A3	H14T3A4	H14T3A5

Jahrgang 2015

F15T1A1	F15T1A2	F15T1A3	F15T1A4	F15T1A5
F15T2A1	F15T2A2	F15T2A3	F15T2A4	F15T2A5
F15T3A1	F15T3A2	F15T3A3	F15T3A4	F15T3A5
H15T1A1	H15T1A2	H15T1A3	H15T1A4	H15T1A5
H15T2A1	H15T2A2	H15T2A3	H15T2A4	H15T2A5
H15T3A1	H15T3A2	H15T3A3	H15T3A4

Jahrgang 2016

F16T1A1	F16T1A2	F16T1A3	F16T1A4	F16T1A5
F16T2A1	F16T2A2	F16T2A3	F16T2A4	F16T2A5
F16T3A1	F16T3A2	F16T3A3	F16T3A4	F16T3A5
H16T1A1	H16T1A2	H16T1A3	H16T1A4	H16T1A5
H16T2A1	H16T2A2	H16T2A3	H16T2A4	H16T2A5
H16T3A1	H16T3A2	H16T3A3	H16T3A4	H16T3A5

Jahrgang 2017

F17T1A1	F17T1A2	F17T1A3	F17T1A4	F17T1A5
F17T2A1	F17T2A2	F17T2A3	F17T2A4	F17T2A5
F17T3A1	F17T3A2	F17T3A3	F17T3A4	F17T3A5
H17T1A1	H17T1A2	H17T1A3	H17T1A4	H17T1A5
H17T2A1	H17T2A2	H17T2A3	H17T2A4	H17T2A5
H17T3A1	H17T3A2	H17T3A3	H17T3A4	H17T3A5

Jahrgang 2018

F18T1A1	F18T1A2	F18T1A3	F18T1A4	F18T1A5
F18T2A1	F18T2A2	F18T2A3	F18T2A4	F18T2A5
F18T3A1	F18T3A2	F18T3A3	F18T3A4	F18T3A5
H18T1A1 (H/K)	H18T1A2 (H/K)	H18T1A3 (H/K)	H18T1A4 (H/K)	H18T1A5 (H/K)
H18T2A1 (H/K)	H18T2A2 (H/K)	H18T2A3 (H/K)	H18T2A4 (H/K)	H18T2A5 (H/K)
H18T3A1 (H/K)	H18T3A2 (H/K)	H18T3A3 (H/K)	H18T3A4 (H/K)	H18T3A5 (H/K)

Jahrgang 2019

F19T1A1 (H/K)	F19T1A2 (H/K)	F19T1A3 (H/K)	F19T1A4 (H/K)	F19T1A5 (H/K)
F19T2A1 (H/K)	F19T2A2 (H/K)	F19T2A3 (H/K)	F19T2A4 (H/K)	F19T2A5 (H/K)
F19T3A1 (H/K)	F19T3A2 (H/K)	F19T3A3 (H/K)	F19T3A4 (H/K)	F19T3A5 (H/K)

III. Sortierung der Aufgaben nach Einzelthemen

(A) Lineare Algebra
- Untervektorräume und Dimensionsbegriff
  
  H15T1A2 F18T3A1 (d)
- Lösungsmengen linearer Gleichungssysteme
  
  F18T2A1 (d) F18T3A1 (b) H18T2A2 (H/K)
- Eigenwerte, Eigenvektoren und Diagonalisierbarkeit
  
  F12T3A1 F14T1A4 F15T1A1 (a) H15T3A3 (c) F16T2A5
  H16T2A3 F17T1A5
- Minimalpolyome, Satz von Cayley-Hamilton und Jordansche Normalform
  
  H13T1A3 H14T2A5 F15T1A1 (b) F16T3A1 (a) H16T3A1
  H16T1A2 F17T3A4 H17T2A1
- Euklidische Vektorräume
  
  F16T1A1
- Aufgaben zur Linearen Algebra in Verbindung mit Gruppenoperationen
  
  F12T3A1 (b) H12T2A1 F13T1A5 F15T2A4 (b) F19T1A2 (H/K)
(B) Gruppentheorie
- Zusatzmaterial: Gruppen der Ordnung 2014 und 2015
- Gruppenaxiome
  
  F19T2A1 (H/K)
- Grundbegriffe der Gruppentheorie (Untergruppen, kommutierende Elemente, Normalteiler, Index)
  
  F12T1A1 H13T1A1 (a) F14T2A3 (a),(c) F14T3A1 F15T3A1
  H15T2A1 H15T3A2 H17T3A2 (a) F18T2A5 (a) H18T1A2 (H/K)
  F19T2A2 (H/K) H19T3A2
- Ordnung von Gruppenelementen
  
  F13T2A1 H13T3A3 H14T3A3 F16T3A1 (b) F17T2A2 (a)
  H17T1A3 F18T3A2 H18T1A1(b,c) (H/K)
- Ordnung linearer Gruppen über endlichen Körpern
  
  H12T1A1 (a) F13T1A3 (a)
- Struktur zyklischer Gruppen
  
  F12T2A2 F16T2A2 (c) F16T3A3 (a) F17T2A1
- endlich erzeugte abelsche Gruppen
  
  F12T2A1 H12T3A1 H18T2A3 (b) (H/K) F19T2A1 (c) (H/K)
- Faktorgruppen und Homomorphiesatz
  
  F13T1A3 (b) F13T2A1 F14T2A3 (b) F17T2A3 (c) H17T3A2 (b)
  F18T2A5 (c) H18T2A3 (a) (H/K)
- Gruppenoperationen und Bahngleichung (weitere Aufgaben unter (A) Lineare Algebra)
  
  H12T1A1 (b) H12T3A2 H12T3A3 (b) H13T2A2 H13T2A3 (a)
  H13T3A4 F14T1A1 H14T1A4 H14T2A3 (b) F15T2A4
  H16T2A1 F17T3A2 (b),(c) H17T2A2 F19T3A2 (H/K) H19T3A3
- Sylowsätze
  
  F12T1A2 F13T1A3 (c) H13T1A2 H14T2A4 H14T3A1
  F15T1A3 (a) F15T3A2 H15T1A3 H16T3A3 F17T1A3
  F17T2A1 F17T2A5 H17T3A1 (a) F18T3A3 H18T3A2 (b) (H/K)
  F19T1A1 (d) (H/K) H19T1A4
- innere direkte Produkte
  
  H14T3A1 H15T1A3 F18T3A3
- semidirekte Produkte
  
  F13T3A1 F16T1A3 H16T2A2
- Auflösbarkeit
  
  F15T1A3 (b) F15T3A2 F16T2A1 H16T1A1 H17T3A1 (b)
  H18T3A1 (a) (H/K)
- symmetrische Gruppen
  
  F12T2A1 (b),(c) H12T3A2 H13T2A3 (a) H13T2A5 H13T3A3
  H15T2A2 F17T3A2 (a) F18T3A1 (c) H18T1A1 (b) (H/K) F19T3A5 (H/K)
  H19T1A1 (d) H19T2A4
- Diedergruppen
  
  F12T1A2 H12T3A1 H17T1A3 F18T2A5 (b),(c)

(C) Ringtheorie

Elementare Begriffe der Ringtheorie (Einheit, Nullteiler, erzeugter Teilring, Ideal)

F12T1A3 F12T2A1 (d) F13T3A2 F13T3A3 F14T2A5
H14T1A2 (a) H14T2A1 F15T3A3 F16T2A5 (b) F16T3A4
H16T1A3 F17T3A3 H17T3A3 F18T2A1 (a) F18T3A1 (a)
H18T2A4 (a) (H/K) H18T3A5 (c) (H/K)

Faktorringe und Homomorphiesatz

F12T3A4	H12T2A3	H13T1A1 (d)	H13T1A4	F14T1A3 (b)
F14T2A1	H14T2A4 (b),(c)	F15T1A4	F15T2A2	H15T3A3 (a)
F16T2A2	F16T3A2	H17T1A4	F18T2A3	F18T3A4
H18T3A5 (a,b)(H/K)	F19T2A4 (H/K)	F19T3A1 (H/K)	F19T3A4 (H/K)

Teilringe quadratischer Zahlkörper und euklidische Ringe

F12T2A3 H12T1A4 F13T1A1 (b) H13T1A1 (b) F14T2A2
F14T3A3 H14T3A4 F16T1A4 F16T2A2 F17T1A1
F19T1A3 (H/K)
größter gemeinsamer Teiler, Lemma von Bézout, Hauptidealringe und faktorielle Ringe

F12T2A3 F12T3A4 F14T1A3 (a) H16T1A4 H18T2A1 (H/K)
H19T3A1
Kongruenzen und Rechnen in Restklassenringen

H13T3A5 F14T3A2 (a),(b) H14T2A3 (a) F15T1A2 H15T1A1
H15T2A3 H15T3A1 F16T1A2 F17T1A4 F17T2A2 (b)
F18T1A5 F18T2A1 (b) F18T2A4 F19T1A1 (a) (H/K) F19T2A3 (H/K)
H19T2A1 H19T2A2
Einheitengruppen endlicher Ringe

H14T3A2 F16T2A2 (b),(c) F17T2A2 (c) F17T2A3 F17T3A5
H17T1A1 H17T1A3
Rechenaufgaben zum Chinesischen Restsatz

F12T1A4 H12T1A5 H12T3A4 F14T3A2 (c) H14T1A5
H14T3A2 H16T2A5
Quadratisches Reziprozitätsgesetz

F13T1A1 (a) F14T3A2 (d)
Elementare Zahlentheorie (ohne Kongruenzrechnung)

F15T2A1

Irreduzibilität und Zerlegung von Polynomen

F12T2A4 (a)	H12T3A5	F13T1A2 (a)	F13T3A4	H13T1A5 (a)
H13T2A1 (a)	H13T3A2 (b)	F14T2A4 (a)	F15T2A3	H15T1A4 (a),(b)
F16T2A4 (a)	H17T1A5	H17T2A3 (c)	F18T1A1	F18T1A4 (a)
F18T2A1 (c)	F18T2A2 (a)	H18T1A1 (a) (H/K)	F19T1A1 (b) (H/K)	F19T3A3 (b) (H/K)
H19T1A1 (a),(b)	H19T3A5 (a)

Polynome mit mehrfachen Nullstellen

F13T1A4 F12T3A3 H12T2A2 F19T1A4 (H/K)

(D) Körpertheorie
- Rechnen in algebraischen Erweiterungen
  
  F12T2A4 (c) H13T1A5 (d) H15T3A3 (b) F16T2A4 (b) F16T3A5 (a),(b)
  F17T1A2 (a) F17T2A4 (a) H17T2A4 (a),(b) F18T2A2 (b) H19T1A1 (c)
- Bestimmung von Minimalpolynomen
  
  F15T3A4 H15T2A5 (b) F16T1A5 (a) F16T2A4 (c) F17T1A2 (b),(c)
  H17T3A4 (a) H18T3A2 (a) (H/K) F19T1A5 (H/K)
- Bestimmung des Zerfällungskörpers eines Polynoms
  
  F12T1A5 (a) F12T3A2 (a) H12T3A3 (a) H13T1A1 (c) F16T2A3 (a)
  F16T3A5 (a) H17T2A4 (a) H17T3A4 (b) F18T3A5 (a) F19T1A5 (b) (H/K)
  F19T3A3 (a,c) (H/K)
- Bestimmung von Erweiterungsgraden, Anwendungen der Gradformel
  
  F12T3A2 (b) F13T3A5 F14T3A4 (a) H15T3A4 (a) F16T2A3 (b)
  F16T3A5 (b) H16T2A4 (b) H17T3A5 F18T3A5 (b) F19T2A5 (a) (H/K)
  H19T1A3 H19T2A3
- Beweisaufgaben zu endlichen und algebraischen Körpererweiterungen
  
  F13T2A5 H14T1A3 H15T2A4 H18T3A4 (H/K)
- endliche Körper
  
  F13T2A4 H13T2A1 H13T2A4 F17T1A5 F17T3A5
  H17T2A3 (a),(b) H18T1A3 (H/K) H18T1A5 (H/K) H19T1A5 H19T3A4
- Einheitswurzeln, Kreisteilungspolynome und Kreisteilungskörper
  
  H12T2A3 F13T1A2 (b) F13T3A5 (a) H14T1A1 (b) F15T3A4 (c)
  H15T1A5 (a) F16T1A5 (b) H18T1A1 (d) (H/K) H18T2A4 (H/K) H18T2A5 (H/K)
  H18T3A1 (b) (H/K) H19T3A5 (b)
- Separabilität, Satz vom primitiven Element und Körper positiver Charakteristik
  
  F14T1A5 F14T2A4 (a) F15T2A5 H16T3A2 F17T3A1
  H17T1A2 H18T3A1 (c) (H/K) H18T3A3 (b) (H/K) H19T2A5

(E) Galoistheorie

Nachweis von Galoiserweiterungen

F12T1A5 (b) F12T2A4 (b),(d) F16T3A5 (c),(d) H16T2A4 F18T2A2 (c)
F18T3A5 (c)
Anwendungen des Hauptsatzes der Galoistheorie (und seiner Ergänzungssätze)

F12T1A5 (d),(e) H12T2A4 H13T2A3 (b) F14T1A2 F14T3A4 (b),(c)
H14T1A1 (a) F15T3A5 H15T3A4 (b),(d) H16T2A4 (c),(d) F17T2A5
H17T2A4 (c) H18T2A5 (a,c) (H/K) H18T3A3 (c) (H/K)
Rechnen mit den Elementen einer Galoisgruppe

F12T3A5 H14T2A2 (a),(b) H15T2A5 (b) F18T3A5 (d) H18T2A5 (b) (H/K)

Eigenschaften von Galoisgruppen (zyklisch, abelsch, auflösbar, Element- und Gruppenordnung)

F12T1A5 (c)	F12T2A4 (e),(f)	F13T1A2 (c)	H14T2A2 (c)	F15T1A5 (b)
H16T3A5	H17T2A4 (b)	H17T2A5	H17T3A4 (c)	H18T1A4 (a) (H/K)
H18T3A3 (a) (H/K)	F19T1A5 (c) (H/K)	F19T2A5 (b) (H/K)	H19T1A2	H19T3A5 (b)

Isomorphietyp von Galoisgruppen

F12T3A2 (c),(d) H12T1A3 (d) H12T3A3 (c),(d) H13T1A5 (b),(c) H13T3A1
F14T2A4 H14T3A5 H15T1A4 H15T1A5 H15T2A5
H15T3A4 F16T3A5 (e) H16T1A5 F17T1A2 F18T1A4
H18T3A2 (c) (H/K)
Zusatzmaterial: Isomorphietypen von Galoisgruppen (Überblick)
Konstruktionen mit Zirkel und Lineal

H12T2A5 H13T3A2 (e),(f)

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Inhaltsbereich

Aufgabensammlung zum Staatsexam in Algebra

Fu�zeile

H13T1A3	H14T2A5	F15T1A1 (b)	F16T3A1 (a)	H16T3A1
H16T1A2	F17T3A4	H17T2A1

F12T1A1	H13T1A1 (a)	F14T2A3 (a),(c)	F14T3A1	F15T3A1
H15T2A1	H15T3A2	H17T3A2 (a)	F18T2A5 (a)	H18T1A2 (H/K)
F19T2A2 (H/K)	H19T3A2

F13T1A3 (b)	F13T2A1	F14T2A3 (b)	F17T2A3 (c)	H17T3A2 (b)
F18T2A5 (c)	H18T2A3 (a) (H/K)

H12T1A1 (b)	H12T3A2	H12T3A3 (b)	H13T2A2	H13T2A3 (a)
H13T3A4	F14T1A1	H14T1A4	H14T2A3 (b)	F15T2A4
H16T2A1	F17T3A2 (b),(c)	H17T2A2	F19T3A2 (H/K)	H19T3A3

F12T1A2	F13T1A3 (c)	H13T1A2	H14T2A4	H14T3A1
F15T1A3 (a)	F15T3A2	H15T1A3	H16T3A3	F17T1A3
F17T2A1	F17T2A5	H17T3A1 (a)	F18T3A3	H18T3A2 (b) (H/K)
F19T1A1 (d) (H/K)	H19T1A4

F15T1A3 (b)	F15T3A2	F16T2A1	H16T1A1	H17T3A1 (b)
H18T3A1 (a) (H/K)

F12T2A1 (b),(c)	H12T3A2	H13T2A3 (a)	H13T2A5	H13T3A3
H15T2A2	F17T3A2 (a)	F18T3A1 (c)	H18T1A1 (b) (H/K)	F19T3A5 (H/K)
H19T1A1 (d)	H19T2A4

F12T1A3	F12T2A1 (d)	F13T3A2	F13T3A3	F14T2A5
H14T1A2 (a)	H14T2A1	F15T3A3	F16T2A5 (b)	F16T3A4
H16T1A3	F17T3A3	H17T3A3	F18T2A1 (a)	F18T3A1 (a)
H18T2A4 (a) (H/K)	H18T3A5 (c) (H/K)

F12T2A3	H12T1A4	F13T1A1 (b)	H13T1A1 (b)	F14T2A2
F14T3A3	H14T3A4	F16T1A4	F16T2A2	F17T1A1
F19T1A3 (H/K)

H13T3A5	F14T3A2 (a),(b)	H14T2A3 (a)	F15T1A2	H15T1A1
H15T2A3	H15T3A1	F16T1A2	F17T1A4	F17T2A2 (b)
F18T1A5	F18T2A1 (b)	F18T2A4	F19T1A1 (a) (H/K)	F19T2A3 (H/K)
H19T2A1	H19T2A2

H14T3A2	F16T2A2 (b),(c)	F17T2A2 (c)	F17T2A3	F17T3A5
H17T1A1	H17T1A3

F12T2A4 (c)	H13T1A5 (d)	H15T3A3 (b)	F16T2A4 (b)	F16T3A5 (a),(b)
F17T1A2 (a)	F17T2A4 (a)	H17T2A4 (a),(b)	F18T2A2 (b)	H19T1A1 (c)

F15T3A4	H15T2A5 (b)	F16T1A5 (a)	F16T2A4 (c)	F17T1A2 (b),(c)
H17T3A4 (a)	H18T3A2 (a) (H/K)	F19T1A5 (H/K)

F12T1A5 (a)	F12T3A2 (a)	H12T3A3 (a)	H13T1A1 (c)	F16T2A3 (a)
F16T3A5 (a)	H17T2A4 (a)	H17T3A4 (b)	F18T3A5 (a)	F19T1A5 (b) (H/K)
F19T3A3 (a,c) (H/K)

F12T3A2 (b)	F13T3A5	F14T3A4 (a)	H15T3A4 (a)	F16T2A3 (b)
F16T3A5 (b)	H16T2A4 (b)	H17T3A5	F18T3A5 (b)	F19T2A5 (a) (H/K)
H19T1A3	H19T2A3

H12T2A3	F13T1A2 (b)	F13T3A5 (a)	H14T1A1 (b)	F15T3A4 (c)
H15T1A5 (a)	F16T1A5 (b)	H18T1A1 (d) (H/K)	H18T2A4 (H/K)	H18T2A5 (H/K)
H18T3A1 (b) (H/K)	H19T3A5 (b)

F12T1A5 (b)	F12T2A4 (b),(d)	F16T3A5 (c),(d)	H16T2A4	F18T2A2 (c)
F18T3A5 (c)

F12T1A5 (d),(e)	H12T2A4	H13T2A3 (b)	F14T1A2	F14T3A4 (b),(c)
H14T1A1 (a)	F15T3A5	H15T3A4 (b),(d)	H16T2A4 (c),(d)	F17T2A5
H17T2A4 (c)	H18T2A5 (a,c) (H/K)	H18T3A3 (c) (H/K)

F12T3A2 (c),(d)	H12T1A3 (d)	H12T3A3 (c),(d)	H13T1A5 (b),(c)	H13T3A1
F14T2A4	H14T3A5	H15T1A4	H15T1A5	H15T2A5
H15T3A4	F16T3A5 (e)	H16T1A5	F17T1A2	F18T1A4
H18T3A2 (c) (H/K)