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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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Wed 03 Jan
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History of solving polynomial equations. The complex numbers and
complex conjugation.
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Fri 05 Jan
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Field Extensions. Ring theory reminders. Irreducible
polynomials and ideals in polynomial rings.
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DF 9.2
FT 7-11
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2
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Mon 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.
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DF 9.2-9.5
FT 11-14
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Problem Set #1
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Wed 10 Jan
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Finitely generated extensions.
Simple extensions. Classification of simple extensions.
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DF 13.1
FT 16-18
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Fri 12 Jan
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Gauss's Lemma and primitive polynomials.
Transcendental and algebraic elements. Minimal polynomial.
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DF 9.1, 13.2
FT 11-13, 18-21
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3
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Mon 15 Jan
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No class: Martin Luther King, Jr. Day
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Problem Set #2
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Wed 17 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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Thu 18 Jan
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X-hour lecture:
Compass and straightedge constructions. Constructible numbers form a field.
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DF 13.3
FT 22-24
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Fri 19 Jan
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Quadratic closure. Construction impossibility proofs.
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DF 13.3
FT 22-24
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4
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Mon 22 Jan
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Splitting fields.
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DF 13.4
FT 28-29
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Problem Set #3
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Wed 24 Jan
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Separability.
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DF 13.5
FT 31-33
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Thu 25 Jan
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X-hour lecture:
Algebraic closure.
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DF 13.4
FT 25-26, 87-90
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Fri 26 Jan
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Frobenius. Perfect fields. Finite fields.
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DF 13.5
FT 27-33
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5
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Mon 29 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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Problem Set #4
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Wed 31 Jan
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Galois extensions.
Linear independence of embeddings.
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DF 14.1-14.2
FT 36-37
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Fri 02 Feb
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Fundamental theorem of Galois theory.
Examples of the Galois correspondence.
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DF 14.2
FT 36-39
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6
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Mon 05 Feb
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Proof of the Galois correspondence.
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DF 14.2
FT 38-41
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Problem Set #5
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Wed 07 Feb
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Proof of the Galois correspondence (continued). Galois group of a
polynomial. Normal subgroups of the Galois group.
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DF 14.2
FT 35-45
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Fri 09 Feb
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Normality. Normal closure. Galois is normal and separable.
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DF 14.2
FT 35-43
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7
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Mon 12 Feb
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Primitive element theorem (algorithmic and set-theoretic).
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DF 14.4
FT 61-63
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Takehome Midterm Exam
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Wed 14 Feb
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Cyclotomic fields. Galois theory of finite fields.
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DF 13.6, 14.3, 14.5
FT 64-67, 53-55
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Fri 16 Feb
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No class!
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8
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Mon 19 Feb
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Radical extensions. Solvability by radicals.
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DF 14.7
FT 45-46, 76-77
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Problem Set #6
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Wed 21 Feb
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Solvability by radicals.
Galois's solvability theorem.
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DF 14.7
FT 45-46, 76-77
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Thu 25 Jan
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X-hour lecture:
Discriminant. Galois perspective on quadratic and cubic extensions.
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DF 14.6
FT 47-49
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Fri 23 Feb
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Quartic extensions and the cubic resolvent.
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DF 14.7
FT 49-52
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9
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Mon 26 Feb
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Quartic extensions.
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DF 14.7
FT 49-52
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Problem Set #7
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Wed 28 Feb
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Computational algebra. LMFDB.
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Fri 01 Mar
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Infinite Galois theory.
Krull topology. Profinite groups.
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DF 650-652
FT 91-96
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10
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Mon 04 Mar
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Infinite Galois theory.
Looking ahead!
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DF 650-652
FT 91-96
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Final Exam Review
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Fri 08 Mar
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Final Exam!
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