Math 370 Fields and Galois Theory
The official syllabus in pdf form.
The text Fields and Galois Theory (v4.60), by J. S. Milne, will be referred to as FT.
For more practice problems, see Galois
theory through exercises, by Juliusz Brzeziński, available
on-line via SpringerLink
Weekly problem sets will be due in class on Thursday.
Weekly Syllabus and Homework
Updated March 7, 2019.
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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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