# MA232: Abstract Algebra II

Unit 4: Galois Theory   Unit 4 concerns one of the most important theories in abstract algebra: Galois theory.  Evariste Galois is one of the most enigmatic figures in mathematics.  Incredibly intelligent, he intimidated many of the finest mathematical minds in France as a teenager.  A proponent of democracy at the height of monarchy in France, the hotheaded Galois was tricked into accepting a duel against a veteran swordsman, Pescheux d’Herbinville, at age 20.  Realizing that he probably would not survive the duel, Galois spent much of his last night alive frantically compiling ideas he had on general solutions of polynomials.  The notes he wrote to his friend Auguste Chevalier became the foundation for what we now know as Galois theory.

First, Galois used permutation groups to show how roots of a polynomial are related.  After learning about permutation groups, we will look at Galois groups, which are sets of automorphisms on an extension field L of some field K.  We will then conclude the unit (and the course) with a long look at the Fundamental Theorem of Galois Theory, which states that subgroups of the Galois group for extension field L for some field K correspond with the subfields of K that contain L.

This unit will take approximately 14 hours to complete.

☐    Subunit 4.1: 4 hours

☐    Subunit 4.2: 5 hours

☐    Subunit 4.3: 5 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Use permutation groups to find solutions to polynomials.
• Determine the Galois group of a polynomial.
• Determine if a polynomial is solvable by radicals by determining if the Galois group of the polynomial is a solvable group.

4.1 Permutation Groups and Solutions of Polynomials   - Lecture: YouTube: “Symmetries of a Star and its Permutation Group” Link: YouTube: “Symmetries of Star and its Permutation Group” (YouTube)

Instructions: Please click on the link and view the video in its entirety.  The video discusses a permutation group of a regular geometric figure (a Star of David). You first viewed this interesting visual presentation in Abstract Algebra I.  Once again, this video may be viewed in a new light due to the material we have covered up to this point.

• Reading: Wolfram MathWorld: “Permutation Group” Link: Wolfram MathWorld: “Permutation Group” (HTML)

Instructions: Please click on the link to read the information on the page.  This webpage contains a concise rendering of information on permutation groups.  It is short, but provides a good explanation in a few sentences.  The page also contains a set of useful links to related topics.

Instructions: Please read the entire article.  This webpage contains a concise rendering of information on permutation groups, including various examples.

Instructions: Please click on the link and view the video in its entirety.  The video discusses general permutations and then discusses permutations in light of groups.  This is a good introduction to the concept.  We saw this video in Abstract Algebra I, but in light of all we have covered since, you may view this video in a new light.

4.2 Galois Groups   - Reading: Wolfram MathWorld: “Galois Group” Link: Wolfram MathWorld: “Galois Group” (HTML)

Instructions: Please click on the link to read the information on the page.  This webpage contains a concise rendering of information on Galois groups.  It is short, but provides a solid explanation in just a few sentences.  The page also contains a set of useful links to related topics.

4.3 The Fundamental Theorem of Galois Theory   - Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Galois Theory” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Galois Theory”

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