Lecture Abstract Algebra



Course Task: In this course we learn about algebraic structures, to get a profound knowledge about groups, rings, fields and basics in Galois theory. It will be connected with number theoretical and geometric applications (dependent on time).


Course ContentsHere is a detailed list of the topics.

* structures (magma, semigroups, monoids, groups)

* groups (subgroups, normal subgroups, Lagrange’s Theorem, homomorphisms)

* examples of groups (symmetric groups, linear groups, free groups),

* factor groups (equivalence relations, partitions, integers mod n)

* rings (unitary rings, intgeral domains, factor rings, Chinese remainder theorem)

* polynomial rings (Hilbert’s basis theorem, properties: factorial, Noethernian)

* Noethernian rings, factorial rings (UFD), Euclidean rings

* fields

* field extensions (algebraic, separable, normal, Galois)

* algebraically closed fields (injective and projective limits)

* Galois theory

* (Non-existence of a radical formula in the coefficients for the roots of a general polynomial of degree greater than four)


Here is the modified syllabus.



  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

   Hint for the solution of Problem 11.1  

Sheet 12 Sheet 13 Sheet 14  Sheet 15*




Final:  The final takes place in the teaching building room 404 on Thursday the 24th of June 2021 from 8:30 to 11:30 am.


Lecture notes: pages 1-160  161-178   179-223   224-264



Shanghaitech University

IMS S506


Every Tuesday and Thursday 13:00-13:45 and 13:55-14:40

Start 14th of September 2021


Office hours: Thursdays 11:30-12:30 IMS S412


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© Daniel Skodlerack