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 January 18, 2025.
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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 07 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 09 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 14 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 16 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 21 Jan
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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 23 Jan
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Splitting fields.
Algebraically closed fields. Algebraic closure.
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DF 13.4
FT 25-26, 28-29, 87-90
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4
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Tue 28 Jan
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Separability.
Frobenius. Perfect fields. Finite fields.
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DF 13.5
FT 27-33
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Thu 30 Jan
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Embeddings. Field automorphisms. Automorphism group.
Constructing automorphisms. Automorphism group calculations.
Fixed fields.
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DF 14.1
FT 27-28, 36-39
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5
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Tue 04 Feb
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Galois extensions.
Linear independence of embeddings.
Fundamental theorem of Galois theory.
Examples of the Galois correspondence.
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DF 14.1-14.2
FT 36-39
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Thu 06 Feb
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Proof of the Galois correspondence.
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DF 14.2
FT 38-41
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6
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Tue 11 Feb
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Galois group of a
polynomial. Normal subgroups of the Galois group.
Normality. Normal closure. Galois is normal and separable.
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DF 14.2
FT 35-45
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Thu 13 Feb
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Primitive element theorem (algorithmic and set-theoretic).
Cyclotomic fields. Galois theory of finite fields.
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DF 13.6, 14.3, 14.4, 14.5
FT 61-63, 64-67, 53-55
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7
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Tue 18 Feb
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Radical extensions. Solvability by radicals.
Solvability by radicals. Galois's solvability theorem.
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DF 14.7
FT 45-46, 76-77
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Thu 20 Feb
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Discriminant. Galois perspective on quadratic and cubic extensions.
Quartic extensions and the cubic resolvent.
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DF 14.6-14.7
FT 47-52
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8
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Tue 25 Feb
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Dedekind's theorem. Computing Galois groups over the rational
numbers. Chebotarev density theorem.
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DF 14.8
FT 55-57
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Thu 27 Feb
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Review of modules. Finitely generated modules. Integral elements and
equivalent conditions. Integral closure (is a ring).
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DF 10.1-10.3, 15.3
ANT 25-28
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Fri 07 Mar
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Integrally closed rings. UFD's are integrally closed. Integral
closure in finite extensions. Integral ring extensions.
Transitivity of integrality. Rings of integers are integrally closed.
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DF 15.3
ANT 28-30
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9
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Tue 04 Mar
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Norm, trace, and discriminant. Integral bases. Rings of integers are
finitely generated. Dedekind domains. Prime decomposition in
Dedekind domains.
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ANT 31-36, 48-49
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Thu 06 Mar
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Explicit description of primes lying over. Galois action on primes.
Decomposition group. Proof of Dedekind's theorem.
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ANT 62-65, 144-146
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10
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Fri 13 Mar
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Final Exam!
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