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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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Mon 06 Jan
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History of solving polynomial equations. The complex numbers.
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Wed 08 Jan
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Review of ring theory: Euclidean domains, PID, UFD. Irreducible
polynomials and ideals in polynomial rings.
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DF 9.2
FT 7-11
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Fri 10 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 11-13
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2
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Mon 13 Jan
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Gauss's Lemma and
primitive polynomials.
Tower law for degrees.
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DF 9.3, 13.1
FT 13-17
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Problem Set #1
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Wed 15 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 13.2
FT 15-19
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Fri 17 Jan
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Algebraic extensions, continued. Compass and straightedge. Constructible numbers. Pythagorean closure.
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DF 13.3
FT 21-23
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3
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Mon 20 Jan
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No class: Martin Luther King, Jr. Day
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Problem Set #2
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Wed 22 Jan
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Construction impossibility proofs, continued. Regular n-gons. Splitting fields.
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DF 13.3-4
FT 28-30
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Fri 24 Jan
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Splitting fields.
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DF 13.4
FT 24-25
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4
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Mon 27 Jan
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Separability.
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DF 13.5
FT 30-33
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Problem Set #3
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Wed 29 Jan
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Multiple roots. Embeddings.
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DF 13.5
FT 30-37
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Thu 30 Jan
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X-hour lecture: Algebraic closure.
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DF 13.4
FT 24-25
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Fri 31 Jan
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Field automorphisms. Automorphism group.
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DF 14.1
FT 36-39
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5
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Mon 03 Feb
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Constructing automorphisms. Automorphism group calculations.
Fixed fields.
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DF 14.1
FT 36-39
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Problem Set #4
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Wed 05 Feb
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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 07 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 10 Feb
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End of proof of the Galois correspondence.
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DF 14.2
FT 38-40
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Problem Set #5
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Wed 12 Feb
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Normality.
Finite fields. Cyclotomic fields.
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DF 14.2-3
FT 37-48
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Fri 14 Feb
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Examples of Galois extensions. Computing Galois groups.
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DF 14.2
FT 52-54
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7
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Mon 17 Feb
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Applications of the Galois correspondence.
Radical extensions. Solvability by radicals.
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FT 4
pp. 42-45
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Takehome Midterm Exam
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Wed 19 Feb
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Radical extensions. Solvability by radicals.
Galois's solvability theorem.
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DF 14.7
FT 42-45, 74-75
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Fri 20 Feb
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Solvability, continued.
Primitive element theorem.
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DF 14.7
FT 59-61, 74-75
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8
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Mon 23 Feb
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Discriminant. Galois perspective on quadratic and cubic extensions.
Quartic extensions.
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DF 14.7
FT 49-51
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Problem Set #6
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Wed 25 Feb
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Quartic extensions.
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DF 14.7
FT 50-58
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9
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Mon 02 Mar
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Finite fields. Cyclotomic fields.
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DF 14.5
FT 62-65
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Problem Set #7
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Wed 04 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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Thu 05 Mar
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Fundamental Theorem revisited.
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DF 650-652
FT 91-96
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Fri 06 Mar
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Taussky-Todd's theorem. Grace Hopper's thesis.
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Notices article
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10
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Mon 09 Mar
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Final Exam!
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Final Exam Review
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