MAT 1101: Algebra II, Winter 2026
Instructor: Florian Herzig;
my last name at math dot toronto dot edu
Office Hours (online/zoom): Fri 3:30-4:30pm or by appointment
TA: Yun-chi Tang
Lectures: Tuesday/Thursday 10:30-12pm
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
The grading scheme will be finalised during the first week of classes.
Homework: 0%
2 Term tests: 30% each
Final: 40%
Term tests: Tue Feb 3 and Tue Mar 10, during class time (10:30am-12:00pm).
There will be no makeup tests! If you miss the test for a valid reason, the grade will be reweighted.
Final: TBA (April)
Homework
Assignments are optional, but you are strongly encouraged to work on them and hand them in for feedback.
I will be using Gradescope (gradescope.ca, not gradescope.com!).
- Assignment 1
- Assignment 2
- Assignment 3
- Assignment 4
- Assignment 5
Rough class schedule
Note that I will be absent during the week of February 9, since I will speak at a conference.
- Jan 6: Galois theory: motivation, field extensions, degree, finite extensions, k-homomorphisms, algebraic and transcendental elements
- Jan 8: tower law, universal property of the field $k_f = k[x]/(f(x))$ (over $k$), algebraic/finitely generated/simple extensions
- Jan 13: algebraic closure (existence)
- Jan 15 (online): algebraic closure (uniqueness), splitting fields, normal extensions
- Jan 20: normal closure, separable extensions
- Jan 22: perfect fields and Frobenius homomorphism, separable degree, tower law for separable degree
- Jan 27: primitive element theorem, Galois extensions, Fundamental Theorem of Galois Theory
- Jan 29: Fundamental Theorem of Galois Theory, example: lattice of subfields of $\mathbb Q(\sqrt[4]2, i)$, finite fields
- Feb 3: (Test 1)
- Feb 5: extensions and polynomials solvable by radical, for a finite Galois extension, radical => solvable Galois group
- Feb 10: (no class)
- Feb 12: (no class)
- Feb 17 (online): for a finite Galois extension, radical <= solvable Galois group, unsolvability of some quintic polynomials
- Feb 19 (online): Commutative Algebra, noetherian rings/modules, Hilbert's basis theorem
- Feb 24: finite/finite type/integral ring extensions, integral closure, Zariski's lemma
- Feb 26: weak Nullstellensatz, algebraic subsets and radical ideals, Nullstellensatz
- Mar 3:
- Mar 5:
- Mar 10: (Test 2)
- Mar 12:
- Mar 17:
- Mar 19:
- Mar 24:
- Mar 26:
- Mar 31:
- Apr 2:
Links