MAT 1101: Algebra II, Winter 2024
Instructor: Florian Herzig;
my last name at math dot toronto dot edu
Office Hours (online/zoom): Tue 12-1pm or by appointment
TA: Nischay Reddy
Lectures: Mondays 1-2pm, Wednesdays 10am-12pm
Official syllabus
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Fields: Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.
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Commutative Rings: Noetherian rings, Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties.
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Structure of semisimple algebras, application to representation theory of finite groups.
(We may not get to invariant theory and primary decomposition, for time reasons)
Useful books for reference (I will not really follow any of them)
- Grillet, Abstract algebra (click link for online access; this book might be the closest to my course)
- Dummit and Foote, Abstract Algebra, 3rd ed.
- Jacobson, Basic Algebra, Volumes I and II.
- Lang, Algebra, 3rd ed.
For representation theory:
- Serre, Linear representations of finite groups
- Alperin and Bell, Groups and representations
- Webb, A Course in Finite Group Representation Theory (available online)
Grading scheme
Homework: 25%
Term test: 30%
Final: 45%
There will be 5 homework assignments. Your lowest homework score will not count towards your grade.
Drop deadline: Tue, Feb 20, 2024
Term test: Wed, Feb 14, 10:10am-12:00pm (instead of class).
There will be no makeup test! If you miss the test for a valid reason, the grade will be reweighted as 36% homework and 74% final.
Final: Tue, Apr 16, 2:00-5:00pm
Homework (tentative schedule)
Assignments will be posted and solutions will be collected via Gradescope (gradescope.ca, not gradescope.com!).
- Assignment 1, due Fri Feb 2
- Assignment 2, due Fri Feb 23
- Assignment 3, due Fri Mar 8
- Assignment 4, due Fri Mar 22
- Assignment 5, due Fri Apr 5
Rough class schedule
- Jan 8: Galois theory: motivation, field extensions, degree, finite extensions
- Jan 10: k-homomorphisms, algebraic and transcendental elements, tower law, universal property of the field $k_f = k[x]/(f(x))$
- Jan 15: algebraic/finitely generated/simple extensions, algebraic closure (existence)
- Jan 17: algebraic closure (uniqueness), splitting fields, normal extensions
- Jan 22: normal closure, separable extensions
- Jan 24: perfect fields and Frobenius homomorphism, separable degree, tower law for separable degree
- Jan 29: primitive element theorem, Galois extensions
- Jan 31: Fundamental Theorem of Galois Theory, example: lattice of subfields of $\mathbb Q(\sqrt[4]2, i)$
- Feb 5: finite fields, extensions and polynomials solvable by radical
- Feb 7: for a finite Galois extension, radical <=> solvable Galois group (proved one direction, started other)
- Feb 12: radical <=> solvable Galois group (proved other direction of theorem), unsolvability of some quintic polynomials; brief start on noetherian rings/modules
- Feb 14: Term Test
- Feb 28: Commutative Algebra: noetherian rings/modules, Hilbert basis theorem, finite/finite type/integral ring extensions
- Mar 4: integral closure, Zariski's lemma
- Mar 6: weak Nullstellensatz, algebraic subsets and radical ideals, Nullstellensatz, Zariski topology
- Mar 11: irreducible components, minimal prime ideals, Jacobson radical
- Mar 13: Nakayama's lemma, beginning of (semi)simple modules, Schur's lemma
- Mar 18: semisimple modules and rings, Artin-Wedderburn theorem
- Mar 20: Artin-Wedderburn, also for k-algebras; beginning of representations of finite groups
- Mar 25: basics on representations and characters
- Mar 27: direct sum, tensor product, and dual representation; permutation representations; character theory
- Apr 1: row/column orthogonality, some character tables
- Apr 3: Burnside's $p^aq^b$ theorem, Frobenius reciprocity
Links