Lecture Abstract Algebra

Course TaskThe first objective is to learn about algebraic structures, to get a profound knowledge about groups, rings and modules and basics in Galois theory. It will be connected with number theoretical and geometric applications. Secondly we strengthen the ability to workout and write abstract proofs in a more academic way.

 

Course ContentsThe algebraic behavior of real vector spaces carries over to a much wider class. Replacing the field of real numbers by a (more general) ring provides a theory with a much more diverse picture. Modules, the analogue for vector spaces, do not need to carry a basis anymore. Rings carry a beautiful set of ideals, which one could think about as a generalization of divisors of integers. Analyzing the behavior of ideals is very successful for proving difficult number theoretical problems. The theory of abstract algebra occurs in almost all mathematical directions. In this course we will learn about:

 

groups, rings, ideals, modules , number theoretical applications, Galois theory, localization, local-global principle

 

Literature:

  1. Lang, Serge Algebra, Springer, 2002, ISBN 978-7506271844
  2. Artin, Michael, Algebra, 2nd edition. Pearson, 2011, ISBN 978-0132413770
  3. Dummit, David S. and Foote, Richard M, Abstract Algebra, 3rd edition, John Wiley & Sons, 2003, ISBN 978-1119222910
  4. Frederick M. GoodmanPrentice Hall, Algebra Abstract and concrete; 1st edition (March 1, 1998), ISBN 978-0132839884

 

 

 

Problem sheets: Sheet 1, Sheet 2, Sheet 3, Sheet 4, Sheet 5,   Sheet 6  Sheet 7

Sheet 8  Sheet 9 Sheet 10  Sheet 11   Sheet 12

 

Lecture notes: pages 1-160  161-178

 

 

Shanghaitech University

Teaching building 404

 

Tuesdays and Thursdays 8:15-9:55

Start 23rd of February 2021

 

Office hours: Tuesdays 10-11 IMS413

 

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© Daniel Skodlerack