Perspectives in Mathematical Sciences


The subject for this portion of the course is modules over rings. We show that every (left or right) module over a division ring is free and introduce and classify semi-simple rings. We prove Maschke's theorem that the group ring of a finite group over a field, the characteristic of which does not divide the order of the group, is semi-simple. We study the case of cyclic groups in detail, where the description of the group ring afforded by the theorem is known as the discrete Fourier transform. Finally, we study rings of integers in number fields, where we introduce the ideal class group first considered by Kummer. Here is a more detailed syllabus: Time and place: Tuesday 2:45-4:15 in Science Building 1, Room 509.

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