Math 81/111 Abstract Algebra: Field and Galois Theory
Require textbook (referred to as DF):
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David S. Dummit and Richard M. Foote,
Abstract Algebra, 3rd Edition
List of other useful texts and resources:
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J. S. Milne,
Fields and Galois Theory, (referred to as FT)
available online via Milne's website
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J. S. Milne,
Algebraic Number Theory, (referred to as ANT)
available online via Milne's website
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Juliusz Brzeziński,
Galois theory through exercises,
available on-line via SpringerLink
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Keith Conrad's expository notes
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Serge Lang,
Algebra, Graduate Texts in Mathematics, vol. 211, Third edition, 2005.
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Ian Stewart,
Galois Theory, Third edition, 2003.
Weekly problem sets will be due in class on Friday.
Weekly Syllabus and Homework
Updated February 20, 2026.
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Week
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Date
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Topics
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Reading
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Homework
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1
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Tue 06 Jan
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History of solving polynomial equations. The complex numbers and
complex conjugation. Field Extensions. Ring theory reminders: PID,
UFD, prime and maximal ideals, prime and irreducible elements, fraction fields. Irreducible polynomials and ideals in polynomial rings.
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DF 9.2
FT 7-11
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Thu 08 Jan
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Roots. Fundamental Theorem of Arithmetic. Reduction mod p.
Irreducibility criteria for polynomials. Irreducible polynomials over finite fields. Eisenstein's criterion. Gauss's Lemma and primitive polynomials.
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DF 9.1-9.5
FT 11-14
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2
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Tue 13 Jan
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Finitely generated extensions.
Simple extensions. Classification of simple extensions.
Transcendental and algebraic elements. Minimal polynomial.
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DF 13.1-13.2
FT 16-21
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Problem Set #1
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Thu 15 Jan
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Tower law for degrees.
Algebraic extensions, continued.
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DF 13.2
FT 14-15, 19-20
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3
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Tue 20 Jan
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Finitely generated algeraic extensions. Compass and straightedge constructions. Constructible numbers form a field.
Quadratic closure. Construction impossibility proofs.
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DF 13.3
FT 22-24
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Problem Set #2
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Thu 22 Jan
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Quadratic closure. Construction impossibility proofs.
Splitting fields.
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DF 13.3-13.4
FT 22-24, 27-30
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4
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Tue 27 Jan
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Splitting fields. Embeddings. Counting embeddings.
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DF 13.4
FT 27-30
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Problem Set #3
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Thu 29 Jan
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Separability.
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DF 13.5
FT 30-33
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Fri 30 Jan
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Extra lecture:
Algebraically closed fields. Algebraic closure.
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DF 13.4
FT 25-26, 87-90
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5
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Tue 03 Feb
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Frobenius. Perfect fields. Finite fields.
Field automorphisms. Automorphism group.
Constructing automorphisms. Automorphism group calculations.
Fixed fields.
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DF 14.1-14.2
FT 33-37
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Problem Set #4
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Thu 05 Feb
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Galois extensions.
Linear independence of embeddings.
Fundamental theorem of Galois theory.
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DF 14.1-14.2
FT 37-41
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6
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Tue 10 Feb
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Examples of the Galois correspondence.
Proof of the Galois correspondence.
Galois group of a polynomial. Normal subgroups of the Galois group.
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DF 14.1-14.2
FT 35-41,44-45
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Takehome Midterm Exam
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Thu 12 Feb
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Normality. Normal closure. Galois is normal and separable.
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DF 14.1-14.2
FT 37-39
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7
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Tue 17 Feb
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Cyclotomic fields.
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DF 13.6
FT 62-65
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Problem Set #5
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Thu 19 Feb
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Radical extensions. Solvability by radicals.
Galois's solvability theorem. Abel-Ruffini theorem.
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DF 14.7
FT 45-46, 76-77
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8
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Tue 24 Feb
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Discriminant. Galois perspective on quadratic and cubic extensions.
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DF 14.6-14.7
FT 47-52
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Problem Set #6
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Thu 26 Feb
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Quartic extensions and the cubic resolvent.
Galois theory of finite fields.
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DF 14.6-8
FT 47-57
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9
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Tue 03 Mar
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Dedekind's theorem. Computing Galois groups over the rational numbers. Chebotarev density theorem.
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DF 14.3, 14.7-8
FT 53-57
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Extra Credit
Problem Set #7
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Thu 05 Mar
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Infinite Galois theory.
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FT Chapter 7
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Fri 06 Mar
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Extra lecture:
Primitive element theorem (algorithmic and set-theoretic).
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DF 14.4
FT 61-63
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10
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Tue 10 Mar
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Infinite Galois theory
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FT Chapter 7
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11
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Mon 16 Mar
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Final Exam!
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Final Exam Review
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