Syllabus
What follows is a tentative syllabus for the class, taken
from the Graduate Handbook.
The actual material covered in class can be found
in Google Docs.
What follows is a tentative syllabus for the class, taken
from the Graduate Handbook:
- Linear and multilinear algebra (4 weeks)
- Minimal and characteristic polynomials. The Cayley-Hamilton Theorem.
- Similarity, Jordan normal form and diagonalization.
- Symmetric and antisymmetric bilinear forms, signature and
diagonalization.
- Tensor products (of modules over commutative rings). Symmetric
and exterior algebra (free modules).
HomR(- , -) and
tensor products.
References: Roman, Chapters 8-10 and 14, Lang, Chapters XV and XVI; Dummit and Foote, Chapter 11.
- Rudiments of homological algebra (2 weeks)
- Categories and functors. Products and coproducts. Universal objects,
Free objects. Examples and applications.
- Exact sequences of modules. Injective and projective modules.
HomR(- , -), for R a commutative ring.
Extensions.
References: Lang, Chapter XX; Dummit and Foote, Part V, Chapter 17.
- Representation Theory of Finite Groups (2 weeks)
- Irreducible representations and Schur's Lemma.
- Characters. Orthogonality. Character table. Complete
reducibility for finite groups. Examples.
References: Lang, Chapter XVIII; Dummit and Foote, Part VI; Serre.
- Galois Theory (6 weeks)
- Irreducible polynomials and simple extensions.
- Existence and uniqueness of splitting fields. Application to
construction of finite fields. The Frobenius morphism.
- Extensions: finite, algebraic, normal, Galois,
transcendental.
- Galois polynomial and group. Fundamental theorem of Galois
theory. Fundamental theorem of symmetric functions.
- Solvability of polynomial equations. Cyclotomic
extensions. Ruler and compass constructions.
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