Math 370 Fields and Galois Theory
The official syllabus in pdf form.
The text Galois Theory, 4th Edition, by Ian Stewart, will be referred to as GT.
Weekly problem sets will be due in class on Thursday.
Weekly Syllabus and Homework
Updated April 24, 2018.
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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 16 Jan
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History of solving polynomial equations. The complex numbers.
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GT 1.1-1.4, 2.1-2.3, 3.1
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Thu 18 Jan
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Field extensions. Review of ring theory: Euclidean domains, PID, UFD.
Irreducible polynomials and ideals in polynomial rings.
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GT 3.2-3.3
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2
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Tue 23 Jan
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Euclidean algorithm. Bezout identity. Algorithmic proof that a
polynomial ring is a UFD. Roots of polynomials.
Fundamental Theorem of Arithmetic.
Irreducibility criteria for polynomials over the integers. Reduction
mod p. Eisenstein's criterion.
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GT 3.3-3.6
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Problem Set #1
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Thu 25 Jan
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Gauss's Lemma(s). Primitive polynomials. More irreducibility.
Field extensions.
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GT 3.3-3.6
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3
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Tue 30 Jan
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Irreducible polynomials over finite fields.
More field extensions. Simple extensions.
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GT 3.4-3.6
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Problem Set #2
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Thu 01 Feb
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Classification of simple extensions.
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GT 5
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4
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Tue 06 Feb
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Tower law for degrees. Multiquadratic extensions.
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GT 6
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Problem Set #3
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Thu 08 Feb
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Splitting fields. Structure of finitely generated fields. Compass
and straightedge.
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GT 6,7.
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5
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Tue 13 Feb
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Compass and straightedge. Pythagorean closure.
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GT 7.
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Problem Set #4
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Thu 15 Feb
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Constructible numbers. Construction impossibility proofs.
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GT 7.
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6
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Tue 20 Feb
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Field automorphisms. Galois groups.
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GT 8.
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Midterm exam 1 review
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Thu 22 Feb
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Midterm exam 1
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7
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Tue 27 Feb
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Galois extensions. Galois correspondence (statement and use).
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GT 8, 9, 11.
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Problem Set #5
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Thu 01 Mar
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More Galois correspondence. Galois perspective on quadratic and cubic
extensions. Normality.
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GT 8, 9.1
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8
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Tue 06 Mar
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Normality.
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GT 9.2.
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Problem Set #6
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Thu 08 Mar
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Separability. Derivations.
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GT 9.3, 17.5.
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9
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Tue 13 Mar
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Spring Break!
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Thu 15 Mar
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Spring Break!
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10
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Tue 20 Mar
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Spring Break!
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Thu 22 Mar
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Spring Break!
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11
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Tue 27 Mar
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Separable extensions. Extensions of derivations.
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GT 9.3, 17.5.
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Problem Set #7
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Thu 29 Mar
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Linear independence of characters and embeddings.
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GT 10.1.
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12
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Tue 03 Apr
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Bounds on the number of embeddings. First proofs toward the Galois correspondence.
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GT 10.1, 12.1.
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Problem Set #8
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Thu 05 Apr
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More proofs toward the Galois correspondence. Galois is
equivalence to normal and separable.
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GT 12.1.
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13
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Tue 10 Apr
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Midterm exam 2
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Midterm exam 2 review
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Thu 12 Apr
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Finish the proof of the Galois correspondence.
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GT 10.1, 12.1.
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14
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Tue 17 Apr
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Primitive element theorem. Radical extensions. Cardano's formula.
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GT 15.1
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Problem Set #9
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Thu 19 Apr
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Solvability by radicals. Radical Galois extensions have solvable group.
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GT 14, 15.1.
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15
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Tue 24 Apr
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Recap of radical Galois extensions. Insolvability of the general
quintic.
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GT 15.1-15.3, exercise 22.7.
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Problem Set #10
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Thu 26 Apr
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Cubic resolvant of a quartic.
Grace Hopper's thesis
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16
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Tue 01 May
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Reading period.
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Final Exam Review
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Thu 03 May
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Reading period
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Sat 05 May
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Final Exam!
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