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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 15 Jan
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History of solving polynomial equations. The complex numbers.
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Thu 17 Jan
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Review of ring theory: Euclidean domains, PID, UFD. Irreducible
polynomials and ideals in polynomial rings. Roots. Fundamental
Theorem of Arithmetic. Reduction
mod p.
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FT 1
pp. 7-11
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2
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Tue 22 Jan
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Gauss's Lemma and
primitive polynomials. Irreducibility criteria for polynomials. Irreducible polynomials over
finite fields. Eisenstein's criterion.
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FT 1
pp. 11-13
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Problem Set #1
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Thu 24 Jan
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Field extensions. Tower law for degrees.
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FT 1
pp. 13-17
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3
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Tue 29 Jan
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Simple extensions. Classification of simple extensions.
Transcendental and algebraic elements. Minimal polynomial.
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FT 1
pp. 16-20
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Problem Set #2
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Thu 31 Jan
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Algebraic simple extensions. Finitely generated extensions.
Algebraic closure.
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FT 1
pp. 16-20
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4
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Tue 05 Feb
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Compass and straightedge. Constructible numbers. Pythagorean closure.
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FT 1
pp. 21-23
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Problem Set #3
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Thu 07 Feb
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Construction impossibility proofs.
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FT 2
pp. 21-23
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5
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Tue 12 Feb
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More construction impossibility proofs. Regular n-gons.
Splitting field.
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FT 2
pp. 27-30
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Problem Set #4
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Thu 14 Feb
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Extension properties. Embeddings.
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FT 2
pp. 27-30
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6
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Tue 19 Feb
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Midterm exam 1
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Midterm exam 1 review
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Thu 21 Feb
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Multiple roots. Separable polynomials. Separable extensions.
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FT 3
pp. 30-37
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7
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Tue 26 Feb
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Field automorphisms. Automorphism group.
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FT 3
pp. 36-39
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Problem Set #5
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Thu 28 Feb
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Constructing automorphisms. Automorphism group calculations.
Fixed fields.
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FT 3
pp. 36-39
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8
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Tue 05 Mar
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Galois extensions.
Linear independence of embeddings.
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FT 3
pp. 36-37
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Problem Set #6
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Thu 07 Mar
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Fundamental theorem of Galois theory.
Examples of the Galois correspondence.
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FT 3
pp.
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9
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Tue 12 Mar
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Spring Break!
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Thu 14 Mar
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Spring Break!
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10
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Tue 19 Mar
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Spring Break!
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Thu 21 Mar
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Spring Break!
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11
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Tue 26 Mar
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End of proof of the Galois correspondence.
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FT 4
pp. 42-45
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Problem Set #7
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Thu 28 Mar
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Sylvester's forumula for the discriminant. Taussky-Todd's theorem. Grace Hopper's thesis.
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Notices article
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12
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Tue 02 Apr
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Normality.
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FT 4
pp. 37-39
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Problem Set #8
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Thu 04 Apr
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Normality and normal subgroups of Galois groups.
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FT 4
pp. 37-39
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13
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Tue 09 Apr
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Midterm exam 2
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Midterm exam 2 review
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Thu 11 Apr
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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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14
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Tue 16 Apr
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Radical extensions. Solvability by radicals.
Galois's solvability theorem.
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FT 4, 5
pp. 42-45, 74-75
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Problem Set #9
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Thu 18 Apr
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Primitive element theorem.
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FT 5
pp. 59-61
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15
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Tue 23 Apr
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Discriminant. Galois perspective on quadratic and cubic extensions.
Quartic extensions.
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FT 4
pp. 49-51
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Problem Set #10
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Thu 25 Apr
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Quartic extensions. Computing Galois groups. Where does algebra go
from here?
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FT 5
pp. 50-58
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16
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Tue 30 Apr
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Reading period.
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Final Exam Review
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Thu 02 May
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Reading period
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Fri 03 May
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Final Exam!
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