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.
Unit 4 Time Advisory
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.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.Reading: Wikipedia’s “Permutations” Link: Wikipedia’s “Permutation Group” (PDF)
Instructions: Please read the entire article. This webpage contains a concise rendering of information on permutation groups, including various examples.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).Lecture: YouTube: “Group Theory Permutations” Link: YouTube: “Group Theory Permutations” (YouTube)
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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
- Reading: Wikipedia's “Galois Group”
Link: Wikipedia’s “Galois
Group” (HTML)
Instructions: Please click on the link above to read the material. This webpage contains useful information about the Galois Group.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).
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”
Also available in:
[iBooks](http://www.saylor.org/site/wp-content/uploads/2011/09/Abstract-Algebra_-Theory-and-Applicatio-Thomas-W.-Judson.epub)
Instructions: Please read 23.2: The Fundamental Theorem, pages 377
– 383.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410 – 417. The material linked above is
licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be
found [here](http://abstract.ups.edu/download.html) (HTML)
Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Galois Theory”: “Exercise Problems 1, 2, 3, and 9” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Galois Theory”: “Exercise Problems 1, 2, 3, and 9”
Also available in:
iBooks
Instructions: Do problems 1, 2, 3, and 9 on page 391. The solution can be found on page 409.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)