Math 314: Galois Theory

Lectures:   Are in no known time slot: Monday 2:30-3:20, Tuesday 10:30-11:20, and Thursday 9:30-10:20, all in Jeff 319.

Textbook   A Course in Galois Theory by D.J.H. Garling.

‘ I hope to interest the Academy in announcing that among the papers of Évariste Galois I have found a solution, as precise as it is profound, of this beautiful problem: whether or not it is soluble by radicals ....’

    — Joseph Liouville addressing the French Acadamy of Sciences on 4 July 1843.

Homework Assignments

Assignment 1:   .ps   .pdf  .dvi
Assignment 2:   .ps   .pdf  .dvi
Assignment 3:   .ps   .pdf  .dvi
Assignment 4:   .ps   .pdf  .dvi

Final Exam: .ps   .pdf  .dvi   due Friday, April 23 on or before 4 pm.

Lecture Topics

The plan for future lectures is flexible -- depending on how things are going in class I'll readjust the topics and the dates.

• Lecture 1 (Monday, January 5): Introduction to the course. Definition of field and some examples.
• Lecture 2 (Tuesday, January 6): More examples of fields. Characteristic of a field, prime field.
• Lecture 3 (Thursday, January 8): Definition of a ring, examples.
• Lecture 4 (Monday , January 12): Integral domains, fraction fields.
• Lecture 5 (Tuesday, January 13): Ring homomorphisms, ideals.
• Lecture 6 (Thursday, January 15): Quotient Rings part I.
• Lecture 7 (Monday, January 19): Field extensions, examples. Degree of a field extension (Guest lecturer: Diane Maclagan).
• Lecture 8 (Tuesday, January 20): Quotient Rings Part II.
• Lecture 9 (Thursday, January 22): Criterion for a ring to be a field. Principal ideal domains.
• Lecture 10 (Monday, January 26): Polynomial rings in one variable over a field. Irreducible polynomials.
• Lecture 11 (Tuesday, January 27): Simple extensions. Algebraic and trancendental elements.
• Lecture 12 (Thursday, January 29): More about simple extensions.
• Lecture 13 (Monday, February 2): Even more about simple extensions.
• Lecture 14 (Tuesday, February 3): Ruler and Compass constructions, Part I.
• Lecture 15 (Thursday, February 5): Field Automorphisms.
• Lecture 16 (Monday, February 9): Bound on size of field automorphisms. Good and Bad extensions.
• Lecture 17 (Tuesday, February 10): Splitting fields, construction.
• Lecture 18 (Thursday, February 12): Normal Extensions. Lifting field homomorphisms.
• Lecture 19 (Monday, February 23): Normal Extensions II: more lifting.
• Lecture 20 (Tuesday, February 24): Separable extensions. Example of a non-separable extension.
• Lecture 21 (Thursday, February 26): Differential criterion for separable extensions. Separability in characteristic 0.
• Lecture 22 (Monday, March 1): Galois extensions. Artin's proposition.
• Lecture 23 (Tuesday, March 2): The Galois correspondence.
• Lecture 24 (Thursday, March 4): Galois correspondence II.
• Lecture 25 (Monday, March 8): A detailed example.
• Lecture 26 (Tuesday, March 9): Detailed example part II.
• Lecture 27 (Thursday, March 11): Finite fields.
• Lecture 28 (Monday, March 15): Finite fields II. The theorem of the primitive element.
• Lecture 29 (Tuesday, March 16): Computation of Galois groups in small degrees.
• Lecture 30 (Thursday, March 18): Computation of Galois groups in a slightly larger degree.
• Lecture 31 (Monday, March 22): Solving polynomial equations.
• Lecture 32 (Tuesday, March 23): Solvable groups. Statement of Galois' theorem.
• Lecture 33 (Thursday, March 25): Cyclotomic extensions.
• Lecture 34 (Monday, March 29): Kummer extensions.
• Lecture 35 (Tuesday, March 30): Proof of Galois' theorem on solvability of polynomial equations.
• Lecture 36 (Thursday, April 1): Clean up lecture: Proof of Artin's proposition and other unresolved arguments.