MAT 535
Syllabus Spring 2021
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Week |
Date |
Topics |
HW(due dates in this table overwrite dates in pdf files) |
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Week
1 |
2/1 |
No classes due to snow |
Problems are taken from Dummitt and
Foote, Abstract algebra |
|
2/3 |
Modules over Principal Ideal Domains |
HWs on multi-linear algebra are based on these notes from last semester |
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Week 2 |
2/8 |
Symmetric bilinear forms, quadratic forms, transformation to the canonical diagonal form, the law of inertia. Positive-definite quadratic forms, Sylvester’s criterion. |
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|
2/10 |
Gauss, Cholesky and Iwasawa decompositions. Skew-symmetric bilinear forms, symplectic basis. The Pfaffian |
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Week 3 |
2/15 |
Free, injective, projective, and flat modules. |
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2/17 |
Short five lemma. The snake lemma. Categories and functors, examples. Products and coproducts. |
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Week 4 |
2/22 |
Universal objects and free objects. Examples and applications. |
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|
2/24 |
Exact sequences of modules. HomR(- , -), for R a commutative ring. |
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Week 5 |
3/1 |
Cochain complexes and long exact sequence in cohomology. Projective resolution of an R-module and derived functors Ext and Tor. |
Problem set 4. p. 914-1,2,4;p. 918-
1,2,3;p. 403- 3–7,20–22; p. 791-1–5, 7, 10, 12–14 |
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3/3 |
The cohomology of groups; example of a finite cyclic group. Cross homomorphisms and H1(G,A); group extensions and H2(G,A). |
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Week 6 |
3/8 |
Field theory: field extensions, algebraic extensions. |
Problem set 5. p. 791 -1,2,4,21; p. 780 -Examples (1)–(2); p. 786 -Examples (1)–(2); p. 801 -Examples
(1)–(2);p. 809 -1,2,8 ;p. 809 -4,5 |
|
3/10 |
Midterm 1 |
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Week 7 |
3/15 |
Splitting fields, algebraic closures and algebraically closed fields. |
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|
3/17 |
Separable and inseparable extensions. |
Problem set 6. p. 519- 2; p. 519- 1,4; p. 530–531 4,8,20,22; p. 529–531 3,9,16,19 |
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Week 8 |
3/22 |
Finite fields, cyclotomic polynomials and extensions. |
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|
3/24 |
The primitive element theorem. Galois theory: basic definitions, examples. |
Problem set 7. p. 545 3,5;p. 551–552-
6,11;p. 555–556-7, 8 |
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Week 9 |
3/29 |
The fundamental theorem of Galois theory. |
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|
3/31 |
Examples. Finite fields. Linear independence of characters. |
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Week 10 |
4/5 |
Hilbert’s Theorem 90. Cyclotomic extensions and
abelian extensions over ℚ. More on Galois
correspondence |
Problem set 8. p. 567-4, 5,8;pp. 581–585-
1, 2, 3, 4, 6, 7, 10, 12, 14, 17, 18 ,31 |
|
4/7 |
Galois groups of polynomials, solvability in
radicals. |
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Week 11 |
4/12 |
Integral extensions and closures, algebraic integers. Dedekind domains. Affine algebraic sets and Hilbert’s Nullstellensatz. |
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4/14 |
Midterm 2 |
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Week 12 |
4/19 |
Representation theory of finite groups, examples, including the regular representation. Irreducible, indecomposable and completely reducible representations. Maschke’s theorem. |
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4/21 |
Wedderburn’s theorem. Basic properties of characters. Schur’s Lemma and orthogonality of characters. Decomposition of the regular representation. |
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Week 13 |
4/26 |
The characters of irreducible representations as an orthonormal basis in the space of central functions.. |
Problem set 11 p.852-5,8,10,24,p864-11,14,p876-2,5 |
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4/28 |
The second orthogonality relation for characters. Character tables, examples |
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Week 14 |
5/3 |
Problem set 12 p.885-2,10,15 |
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5/5 |
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5/11 |
Final Exam: |