Math 4010 Section 103 (Fall 2025) Homepage
Introduction to Galois Theory
News & Announcements
- 11/14
- Final Project & Presentation Guidelines and Suggested Topics:
Instead of a Final Exam, we will have a final "project" of sorts.
Everyone should select a topic somehow related to this class
(I have a few suggestions listed below). This should be something
that we didn't cover or at least something we didn't cover in detail.
- Pick and topic and clear it with me (in person or via email).
- Study your topic.
- Create a handout. This should be at least one page front and back.
- Your classmates are your audience. Build off of background from class.
- Let us know where to go for further reading somewhere in your handout.
[Cite any valuable resources - like websites and textbooks - that you found.]
- You might want to give a brief historical blurb.
- Would worked out examples help us understand your topic? If so, include a few.
- Is your topic about a "big theorem"? Include its proof or a sketch of its proof.
- During the final exam period, everyone will give a brief presentation
of their topic. Our class is small so everyone should have as much time
as they need. Plan on having about 10 minutes.
- Your presentation could use slides (like tex or powerpoint) or you could
just use the board and your handout.
- I can print off copies of your handout for you [if you send your handout to me in time].
My Fall 2020 MAT 2110 page has some sample slides.
Or Slides (.pdf) [Source: (.tex)] from a recent talk I gave.
Handout? Why not model after one of my monstrosities. For example:
Differential Algebra and Liouville's Theorem [Source: (.tex)]
Some differential Galois Theory [Source: (.tex)]
Suggested topics/ideas:
- Symmetric Functions / Classical Galois Theory
- Unsolvablity of a Generic Quintic
- History of Solutions of Cubics, Quartics, etc.
- Infinite Dimensional Issues
- Transcendence Degrees / Algebraic Independence
- Algebraic Integers / Algebraic Number Theory
- Artin-Schreier Theorem
- Artin-Wedderburn Theorem & Semisimple Rings
- Wedderburn's Little Theorem
- Dedekind Rings and Valuations
- Group Actions & Sylow Theorems
- Origami Constructability
- Varieties, Radical Ideals, & Algebraic Geometry
- Elliptic Curves
- Applications like Cryptography
Selected Topics:
- Alexia - Elliptic Curves
- Gavin - Category Theory
- Micah - Constructability
- Sam - Origami Constructability
- Brody - Risch Algorithm
- 11/12
- Upcoming Handouts:
• The Conjugate Trick [Maple source: (.mw)]
• Differential Algebra and Liouville's Theorem [Source: (.tex)]
• Some differential Galois Theory [Source: (.tex)]
- 10/22
-
Homework #7 [Source: (.tex)]
is due Wednesday, October 29th.
Upcoming Handouts:
• The Alternating Group [Source: (.tex)]
• Characters [Source: (.tex)]
• Galois correspondence example [Source: (.tex)]
- 10/15
-
Homework #6 [Source: (.tex)]
is due Wednesday, October 22nd.
Forgotten Handouts:
• Irreducibilty for Polynomials [Source: (.tex)]
• Some Field Theory [Source: (.tex)]
- 10/03
-
Homework #5 [Source: (.tex)]
is due Friday, October 10th.
Handout:
• Transcendental Functions [Source: (.tex)]
- 9/26
-
Test #1 is Monday (September 29th).
To review I recommend looking over our homework answer keys. In addition old Rutgers Exams are
full of good stuff to consider:
• Exam #1 and Answer Key
• Exam #2 and Answer Key
Most of Exam #1 looks like good review. Of course, some of the material is a little off topic.
Also, be aware that some definitions (e.g. gcd is slightly different) & notations (e.g. M(F)
for 2x2 matrices over F) may be a little different from ours. Exam #2 looks good as well.
Generally I would down-play problems about matrices and Exam #2 8(b) is outside the scope
of what we've done.
You can also check out old Math 4720/5210 (Abstract Algebra) exams:
• Test #2
• Final Exam
Test #2 is great stuff to look at. Although there's no answer key - but you can ask me.
Ignore stuff about fields of fractions. Final Exam: 1,2(a)-(f),3,4,5(a),6,9 are relevant.
Again, there's no key - but you can ask me.
- 9/22
-
Test #1 is Monday, September 29th.
(More information to come soon.)
- 9/15
-
Homework #4 [Source: (.tex)]
is due Wednesday, September 24th.
- 9/10
-
Handouts:
• Chinese Remaindering [Source: (.tex)]
• Irreducibilty for Polynomials [Source: (.tex)]
Possibly of interest:
• Partial Fraction Decompositions
- 9/08
-
Homework #3 [Source: (.tex)]
is due Monday, September 15th.
- 8/29
-
Handout:
• Factorization Theory [Source: (.tex)]
In case we care, division with remainders in Maple examples (.mw)
Homework #2 [Source: (.tex)]
is due Monday, September 8th.
- 8/25
-
Handout:
• Isomorphism Theorems [Source: (.tex)]
- 8/21
-
Homework #1 [Source: (.tex)]
is due Friday, August 29th.
- 8/18
-
Handouts:
• Ring Definitions [Source: (.tex)]
• Ring Examples [Source: (.tex)]
- 8/07
-
The syllabus,
tentative schedule, &
ASULearn
have been updated.
For some questionable history about Galois:
Stefanie (The Ballad of Galois).
MAT 4010-103
"Introduction to Galois Theory"
Everyone knows how to solve the quadratic equation. Although much more
involved, there are also corresponding cubic and quartic equations.
However, it turns out that no such "quintic formula" exists! This was
first shown in the 1800's by Abel (and sort of by Ruffini in 1800). In the
1820's Galois developed a complete theory which explains exactly when a
polynomial equation is "solvable".
This course will include a review of basic group and ring theory. We will
cover some of the theory of vector spaces (linear algebra) and then study
field extensions. This will then allow us to show the impossibility of
obtaining certain compass-straightedge geometric constructions (eg.
trisecting angles, doubling cubes, squaring circles). Next, we will
develop the Galois correspondence relating field extensions to (Galois)
groups of automorphisms and then show the impossibility of solving certain
quintic equations.
If there is time, I also plan on sketching out some of "differential Galois
theory" and discuss solvability of differential equations.
Prerequisite details: MAT 3110. Although I will assume everyone has had an
introductory course in modern algebra and linear algebra, I will take time to
review necessary background material. I will assume that students are familiar
with the definition of a ring, but not that they remember any substantial ring theory.
Our main text will be "Galois Theory" (2nd edition) by Joseph Rotman
For more details about "What is Galois theory?" I recommend the wikipedia
article... https://en.wikipedia.org/wiki/Galois_theory ...or come talk to me!
If you have any questions specifically about this course, please feel free to contact Dr. Bill Cook at: cookwj@appstate.edu