tel: 736-2100 Ext 33968
e-mail: bergeron@mathstat.yorku.ca
Office: 2029 TEL BUILDING.
Books
Algebra, T. W. Hungerford, GTM Springer. (recommended but not required) Abstract Algebra, Dummit and Foote, Willey. (Highly recommended but not required) An Introduction to Computational Algebraic Geometry and Commutative Algebra, D. A. Cox,
J. Little and D. O'shea, UTM Springer (Highly recommended but not required)
I plan as follow:
[I have add links to notes from John Campbell[JC] taken in 2017 when available]
Jordan canonical form
[Exercise: Show that: T,T': V --> V are two similar linear
transformations iff (V,T) and (V,T') are isomorphic F[x]-modules. ]
Further ring and module theory:
[ see D&F 10.5]
injective and projective modules
I plan to be done with R-module before reading week.
We should have a midterm returning from that week.
The midterm should cover categories and R-modules.
Fields theory and Galois theorems:
[ see D&F all Chap. 13 and Chap. 14 Sec. 1--8 ]
Fields extensions
splitting fields
automorphism group of Fields
Galois correspondence
Galois groups of polynomials
solving polynomials with radicals
IF TIME ALLOW.
Further ring and module theory (We may skip this?):
semisimple ring and Wedderburn Theorem
Introduction to algebraic geometry:
Varieties
radical ideals
Hilbert's nullstellensatz
More on Categories
Proposed Presentation or Projects (let me know in advance your intent so that we can coordinate when to present it and if it is relevant):
My list is not inclusive and is just suggestions
Pick a Theorem above and present it in class (let me know in advance to coordinate when to present it)
Wedderburn Theorem
Hilbert's nullstellensatz
Newton's Theorem on Symmetric functions.
...
Program a canonical form
Compute discriminant of any degree
Propose a relevant topic or project.
Evaluation:
Students will be evaluated on five aspects (which are parts of the life of any living mathematician). The final grade will be base on the average of the best three of the following four:.
1 Project/Homework (working on an extended project or working on exercises)
2 Midterms (Writing exams).
3 Oral Presentation (Presenting some special topic or long proofs).
4 Comprehensive exam (writing the comprehensive exam at the end, Note that for some of you it is one of your Ph. D. requirements).
5 Participation in class (Being there, asking questions, being curious, etc.) is ALSO an important aspect of the evaluation. It may help increase any of your average above by up to 10%.
Nantel Bergeron
Office: 2029 TEL Building
email address: bergeron at yorku dot ca
Department of Mathematics and Statistics.
2029 TEL Building
York University
North York, Ontario M3J 1P3, Canada
To
Department's Public Page