(MATH 504) | GSÜ - ECTS Information Guide

Content

Objective

Understanding the fundamental algebraic concepts and being able to use them within professional mathematics.

Course Content

The course has two parts. The first builds the general vocabulary of algebra: structures defined by their laws, quotients through universal properties, what passes from A to A[X], group actions, modules, tensor products. The second treats field extensions and Galois theory: algebraic closure, separability, the Galois correspondence, finite fields and cyclotomy, solvability by radicals. One result ties the parts together: the Hilbert basis theorem, proved for rings, is what the Nullstellensatz needs. Exercises belong to the exposition and are used by later results.

Course Learning Outcomes

1. Algebraic structures and universal properties: to abstract and analyse algebraic systems (groups, rings, modules, algebras) through morphisms, quotients and universal properties. 2. Symmetry, actions and invariants: to apply group actions and invariants in order to characterise the internal symmetries of mathematical objects and to obtain structural classifications. 3. Module theory and multilinear algebra: to unify linear structures within the framework of modules over principal ideal domains; to handle scalar and base change by means of the tensor product. 4. Field extensions and Galois duality: to relate field extensions and polynomial roots to automorphism groups through the Galois correspondence, and thereby to test algebraic solvability.

Teaching and Learning Methods

interactive lectures and exercise sheets

References

Serge Lang, Algebra; gitlab.com/onayg/mat504

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Algebraic structures; quotients; congruences.
2	Noetherian rings and the Hilbert basis theorem; factorial rings and Gauss's theorem.
3	Group actions; orbit-stabiliser; class equation; Cauchy; Burnside.
4	Symmetric group; cycles; signature; simplicity of A_n for n ≥ 5.
5	Sylow's theorems; groups of small order.
6	Modules; Z-modules; K[T]-modules; free modules.
7	Torsion K[T]-modules; Jordan and rational forms; Cayley-Hamilton.
8	Tensor products; universal property; base change.
9	Field extensions; minimal polynomials; Zariski's lemma; Nullstellensatz.
10	Splitting fields; algebraic closure; extension of embeddings.
11	Separability; reduced base change; perfect fields; characteristic p.
12	Normal extensions; primitive element; Galois correspondence.
13	Finite fields; cyclotomic polynomials; Galois group (Z/nZ)^×.
14	Solvable groups; radical extensions; insolvability of the quintic.

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	1	50
Contribution of final exam to overall grade	1	50
Total	2	100

In-Term Studies
Activities	Number	Contribution
Assignments	1	50
Presentation	0	0
Midterm Examinations (including preparation)	0	0
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	0	0
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	1	50

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1		X
2
3						X
4					X
5						X
6					X
7		X
8				X
9		X
10		X
11						X
12			X
13				X
14			X
15		X
16			X
17		X

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	3	42
Working Hours out of Class	0	0	0
Assignments	7	4	28
Presentation	0	0	0
Midterm Examinations (including preparation)	2	35	70
Project	0	0	0
Laboratory	0	0	0
Other Applications	0	0	0
Final Examinations (including preparation)	0	0	0
Quiz	1	40	40
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	0	0	0
Yıl Sonu	0	0	0
Hazırlık Yıl Sonu	0	0	0
Hazırlık Bütünleme	0	0	0
Total Workload			180
Total Workload / 25			7.20
Credits ECTS			7