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 Contents：Here 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.

Literature:

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