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Abstrakte Galois-Theorie: Gruppen, Ringe, Körper by Marc Nieper-Wißkirchen

Abstrakte Galois-Theorie: Gruppen, Ringe, Körper

by Marc Nieper-Wißkirchen
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Book cover type: Paperback
  • ISBN13: 9783662639689
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer Spektrum
  • Publisher Imprint: Springer Spektrum
  • Publication Date:
  • Pages: 254
  • Original Price: EUR 32.7
  • Language: German
  • Edition: 1. Aufl. 2021
  • Item Weight: 440 grams
  • BISAC Subject(s): Algebra / General

In diesem Lehrbuch wird die moderne, abstrakte Sichtweise auf die galoissche Theorie einsteigertauglich dargestellt. Dazu werden die fundamentalen begrifflichen Abstraktionen der Algebra - Gruppe, Ring und Körper - definiert, erklärt und durch Bezüge zu anderen Bereichen der Mathematik sowie der alltäglichen Anschauung mit Leben gefüllt. Gleichzeitig wird der Anwendungsbereich der elementaren Galois-Theorie erweitert, zum Beispiel von Körpererweiterungen über den rationalen Zahlen zu Körpererweiterungen über beliebigen Körpern.

Der Zugang ist konsequent konstruktiv: anstelle von abstrakten Existenzbeweisen treten konstruktive Verfahren, Lösungen algorithmisch zu berechnen. Dies führt zu vielfältigeren Anwendungsmöglichkeiten der Ergebnisse sowie zu einer konzeptionellen Klarheit und einer schärferen Sicht auf die Zusammenhänge. Der Leser wird dadurch an moderne Entwicklungen herangeführt, etwa in der Topostheorie oder der (Homotopie-)Typentheorie, wo eine Sensibilität für konstruktive Schlussweisen zentral ist.

Der Inhalt knüpft an das Lehrbuch "Elementare Galois-Theorie" desselben Autors an, kann aber auch aufbauend auf eher klassischen Einführungen in die Algebra gewinnbringend gelesen werden. Es eignet sich somit gut für eine weiterführende Bachelor-Vorlesung in Algebra, ist aber auch zum vertiefenden Selbststudium gut geeignet.

Die zentralen Aussagen werden bereits innerhalb des Textes zusammenfassend und prägnant dargestellt, der Leser wird so zum Innehalten und Reflektieren angeregt und kann Inhalte gezielt wiederholen. Darüber hinaus gibt es am Ende jedes Kapitels eine Kurzzusammenfassung, mit der noch einmal Schritt für Schritt die wesentlichen Argumente nachvollzogen werden können, sowie zahlreiche Übungsaufgaben mit ansteigendem Schwierigkeitsgrad.


Marc Nieper-Wißkirchen ist Lehrstuhlinhaber für Algebra und Zahlentheorie an der Universität Augsburg. Sein besonderes Interesse gilt der Algebraischen Geometrie, der abstrakten Algebra und der Kategorientheorie, und der damit verbundenen konstruktiven Sichtweise auf die Algebra.

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