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
* 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)
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.
Prof. Daniel Skodllerack
Institute for Mathematical Sciences, Room S413
Mondays 19:00-20:00 and Sundays 19:00-20:00
Email address: dskodlerack at shanghaitech.edu.cn