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Algebra

Algebra


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About the Book

Inhaltsverzeichnis1 Motivation und Voraussetzungen1.1 Ziele1.2 Voraussetzungen2 Körpererweiterungen und algebraische Elemente2.1 Körpererweiterungen2.2 Zwischenkörper und algebraische ElementeAufgaben3 Gruppen3.1 Allgemeine Definition und Folgerungen3.2 Untergruppen und GruppenhomomorphismenAufgaben4 Gruppenquotienten und Normalteiler4.1 Äquivalenzrelationen4.2 Gruppenquotienten4.3 Der Satz von Lagrange4.4 Normalteiler und Faktorgruppen4.5 Der Homomorphiesatz für Gruppen4.6 Endliche zyklische GruppenAufgaben5 Ringe und Ideale5.1 Kommutative Ringe mit Eins5.2 Ringhomomorphismen5.3 Einheiten und Nullteiler5.4 Ideale, Faktorringe und der Homomorphiesatz5.5 Primideale und maximale Ideale5.6 Der chinesische Restsatz5.7 Beispiele von Ringen in quadratischen ZahlkörpernAufgaben6 Euklidische Ringe, Hauptidealringe, Noethersche Ringe6.1 Euklidische Ringe6.2 Der euklidische Algorithmus6.3 Noethersche RingeAufgaben7 Faktorielle Ringe7.1 Primelemente und irreduzible Elemente, faktorielle Ringe7.2 EigenschaftenAufgaben8 Quotientenkörper für IntegritätsbereicheAufgaben9 Irreduzible Polynome in faktoriellen Ringen9.1 Inhalt von Polynomen9.2 Reduktion modulo Primelement9.3 Das Gauß Lemma9.4 Anwendung der Reduktion mod ? und Satz von GaußAufgaben 10 Galoistheorie (I) - Satz A und seine Variante A'10.1 Die wundersame Körperschaffung10.2 Der Zerfällungskörper10.3 Der Satz A und A'10.4 Anwendung im Körperturm 10.5 Die GaloisgruppeAufgaben11 Intermezzo - explizites Beispiel ^5 ���� 777^, 7Aufgaben12 Normale Körpererweiterungen12.1 Algebraischer Abschluss12.2 Fortsetzung von Körperhomomorphismen12.3 Normale ErweiterungenAufgaben13 Separabilität 13.1 Motivation und Definition13.2 Formale Ableitung13.3 Charakteristik eines Körpers und Separabilität13.4 Der Separabilitätsgrad13.5 Der Satz vom primitiven ElementAufgaben14 Galoistheorie (II) - der Hauptsatz14.1 Der Hauptsatz - Statement14.2 Ausblick auf eine Anwendung - Mitternachtsformel für alle Grade?14.3 Beweis des Hauptsatzes14.4 Beweis des ZusatzesAufgaben15 Kreisteilungskörper15.1 Einheitswurzeln15.2 Kreisteilungskörper und -polynomeAufgaben 16 Endliche Körper16.1 Primkörper, endliche Körper und der Frobenius16.2 Endliche KörperAufgaben17 Mehr Gruppentheorie - Gruppenoperationen und Sylow17.1 Gruppenoperationen17.2 Die Sylowsätze17.3 Anwendungen der Sylowsätze und übliche Tricks17.4 Beweis der SylowsätzeAufgaben18 Auflösbarkeit von Polynomgleichungen18.1 Auflösbare Gruppen18.2 Auflösung von Polynomgleichungen durch Radikale18.3 Die allgemeine Gleichung =-ten GradesAufgabenA Beweis der Existenz eines algebraischen Abschlusses B Tricks und Methoden, um Gruppen einer vorgegebenen Ordnung zu klassifizierenB.1 Standardargumente und BeispieleB.2 Explizite BeispieleSachverzeichnis
About the Author: Prof. ​Dr. Marco Hien war nach einem Postdoc-Jahr an der University of Chicago zunächst an der Universität Regensburg tätig. Seit 2010 ist er Professor für Algebra und Zahlentheorie an der Universität Augsburg mit den Forschungsgebieten Algebraische Geometrie und algebraische Analysis. 2020 erhielt er den "Preis für gute Lehre" des Wissenschaftsministeriums Bayern.


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Product Details
  • ISBN-13: 9783662637777
  • Publisher: Springer
  • Publisher Imprint: Springer Spektrum
  • Height: 234 mm
  • No of Pages: 330
  • Spine Width: 19 mm
  • Weight: 562 gr
  • ISBN-10: 3662637774
  • Publisher Date: 02 Nov 2021
  • Binding: Paperback
  • Language: German
  • Returnable: Y
  • Sub Title: Geeignet Zum Selbststudium Oder Für Online-Vorlesungen
  • Width: 156 mm


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