Alex Lopez' office hours are:
Galois Theory is a surprising connection between two seemingly different algebraic theories: the theory of fields, and group theory. It is a beautiful and fundamental theory that allows problems about equations to be translated into problems about groups.
The text will be Dummit and Foote Abstract Algebra, Third edition. The relevant chapters are 13 and 14, but some important background is in Chapters 7, 8 and 9. From Chapter 7 we need the notion of a quotient ring (Section 7.3) and the field of fractions (Section 7.5). Also important are Sections 8.2 on Principal Ideal Domains and Eisenstein's criterion from Section 9.4. Some of this material is covered in Math 120 but we will review it.
There will be two Gradescope Midterms, probably in weeks 4 and 8. There will also be a final.
Midterm 1 will be a timed Gradescope midterm. You will have one hour to do it, plus some extra time for uploading. It will be open Dummit and Foote. You may choose any hour between Wednesday January 31 noon and Saturday Feb 3 noon to do the midterm. There will be a Canvas announcement with further details.
| Week | Sections in Dummit and Foote | ||
| Week 1 (Jan 8) |
| ||
| Week 2 (Jan 17) | Sections 13.1,13.2,13.4 | ||
| Week 3 (Jan 22) | Sections 13.4,13.5 | ||
| Week 4 (Jan 29) | Sections 13.5,13.6,14.1 | ||
| Week 5 (Feb 5) | Sections 14.1,14.2,14.9* | ||
| Week 6 (Feb 12) | Sections 14.1,14.2,14.3,Separable Degree | ||
| Week 7 (Feb 21) | Sections 14.2,14.3,14.4 | ||
| Week 8 (Feb 26) | Sections 14.5, 3.4 (solvability), 14.6 | ||
| Week 9 (Mar 3) | Section 14.7 |
* I will be introducing the notion of separable degree $[E:F]_s$ early. In the book this is in Section 14.9 but it is useful to have it earlier. So for Week 5 I am listing Section 14.9 for this concept only.
Here are some notes about separable degree, including the useful definition and the result $[E:F]_s = [E:K]_s[K:F]_s$ which I proved in class on Feb 7 and 9.
We skipped Section 13.3 on ruler and compass constructions. This is an interesting application but not needed for anything else. Since we only have 10 weeks, we'll skip it.
During Week 1 we will review some material from Chapters 7, 8 and 9 of Dummit and Foote. Most of this is review of Math 120. These facts are as follows.
From Chapter 8, we will need the fact that if $F$ is a field, the polynomial ring $F[X]$ is a principal ideal domain. This is because it is a Euclidean domain (Proposition 1 in Section 8.1 and Example 2 on page 271).
From Section 7.3 we will need the quotient ring and the first isomorphism theorem (Section 7.3, Theorem 7). From Section 7.5, we need to know that if $R$ is an integral domain then it can be embedded in a field. In fact, there is a smallest field containing $R$, the Field of Fractions defined in Section 7.4.
In Chapter 9, there are some useful facts, such as Gauss's Lemma: if $R$ is a unique factorization domain and $F$ its field of fractions, and $f\in R[X]$ then $f$ is irreduccible in $R[X]$ if and only if it is irreducible in $F[X]$. An application of this is Eisenstein's criterion which is useful in many cases for checking that polynomials are irreducible.
In Chapter 3, Section 3.4 was skipped in Math 120 (I am told) but we will want to review the notion of solvable groups. Thus one HW problem from this section was included in HW7, and we'll discuss this notion before we discuss the problem of solvability by radicals in Section~14.7.
Homework will be posted here and also on Gradescope. Generally homeworks will be due in Gradescope on Tuesdays. However the first HW will be due on Thursday of Week 2.
| Homework | Dummit and Foote | Solutions | ||||
| HW1 (due Thursday Jan 18, 2024) |
| Solutions to HW1 | ||||
| HW2 (due Tuesday Jan 23, 2024) |
| Solutions to HW2 | ||||
| HW3 (due Tuesday Jan 30, 2024) |
| Solutions to HW3 | ||||
| HW4 (due Tuesday February 6, 2024) |
| Solutions to HW4 | ||||
| HW5 (due Tuesday February 13, 2024) |
| Solutions to HW5 | ||||
| HW6 (due Wednesday February 21, 2024) |
| Solutions to HW6 | ||||
| HW7 (due Wednesday February 28, 2024) |
| Solutions to HW7 | ||||
| HW8 (due Thursday, March 7, 2024) |
| Solutions to HW8 |
Beginning in Week 8 we will be starting Dummit and Foote Section 14.7. Historically the question of when equations are solvable by radicals motivated a great deal of what became Galois theory. Here are links to the wikipedia pages of the Italian mathematicians who investigated the solution of cubic and quartic equations, and the invention of the complex numbers.