Math 121: Galois Theory (Winter 2017)

This class will meet 1:30 PM on Tuesdays and Thursdays in Room 305, Building 200.

The text will be Dummit and Foote Abstract Algebra, Third edition. The relevant chapters are 13 and 14, but important background is in Chapters 8 and 9. In addition, I can recommend the following sources.

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.

Both texts are available on-line through the Stanford libraries.

Administrative Details

Office hours

I will have office hours M,Tu,W,Th at 12 noon. You can see me other times either by appointment or if I am in my office and not otherwise busy.

Midterm

The midterm will be in class on Thursday February 9. It will cover Chapter 13 (except Section 13.3 on ruler and compass constructions will not be covered; also Eisenstein's criterion and Gauss' Lemma from Chapter 9, and Section 14.1.

Final

The Final is scheduled March 20 at 12:15 in Room 200-305. This is our usual classroom.

Homeworks

If you submit your homeworks by email, please observe the following.

HomeworkDummit and FooteSolutions
HW1 (due January 18)
13.1 # 1,4,5 (p.519);
13.2 # 1,3,7,8,14 (p.529)
HW1 Solutions
HW2 (due January 25)
13.2 # 16,19,21 (p.529);
13.4 # 1,2,3,5 (p.545)
HW2 Solutions
HW3 (due February 1)
13.4 # 6, (p.545);
13.5 # 1,2,5,6,8 (p.551);
13.6 # 1,10 (p.555);
HW3 Solutions
HW4 (due February 8)
9.4 # 2(a,c),7 (p.311);
13.6 # 7 (p.555);
14.1 # 1,2,3,4 (p.566);
14.2 # 1 (p.581);
HW4 Solutions
HW5 (due February 15)
14.2 # 5,6,13 (p.581);
14.3 # 5,8 (p.589);
HW5 Solutions
HW6 (due February 22)
14.2 # 17, 18, 21, 22 (p.581);
14.3 # 9,10 (p.589);
14.5 # 3,4 (p.603);
HW6 Solutions
March 1: Project due (March 8 will be last HW)
HW7 (due March 8)
3.4 # 5, 12 (p.106);
5.5 #8 (p.184);
14.6 # 1,2,3,4,44,49 (p.617);
HW7 Solutions

Project

The Project was assigned Wednesday February 15 and due Wednesday March 1.

The purpose of the project is to use Galois theory to prove the quadratic reciprocity law, a cornerstone of number theory. I will grade this myself. It will serve instead of a second midterm.

Rules: You may discuss this project with each other, or with me. You may not show each other your written work. If you make use of resources outside the class, let me know what they are.

Biographies for the Theory of Equations

Notes

Note 4: Separable degree

The notion of separable degree is not used in Dummit and Foote but it is important and clarifies the structure of the proofs. You may find it discussed in Lang's Algebra, Section V.4 beginning on page 239. Let $E/F$ be a finite field extension. The separable degree $[E:F]_s$ equals the number of different embeddings of $E$ over $F$ into any sufficiently large extension $\Omega/F$. Then Lang proves:

Proposition 1. If $E\supseteq K\supseteq F$ then \[[E:F]_s = [E:K]_s[K:F]_s.\]

Proposition 2. If $E=F(\alpha)$ where $\alpha$ is a root of an irreducible polynomial $f\in F[x]$, then $[E:F]_s$ is the number of distinct roots of $f$. On the other hand, $[E:F]$ is the degree of $f$. So $[E:F]_s\leqslant [E:F]$ with equality if and only if $f$ is separable.

As a consequence of these facts, let us check the equivalence of two definitions of Galois. Dummit and Foote define $E/F$ to be Galois if $|\hbox{Aut}(E/F)|=[E:F]$. On the other hand, Lang defined $E/F$ to be Galois if it is normal and separable. To see these two notions are equivalent, note that every automorphism of $E/F$ is an embedding of $E$ into a splitting field $\Omega$, so it follows from the definition that \[|\hbox{Aut}(E/F)|\leqslant [E:F]_s\] with equality if and only if $E/F$ is normal. On the other hand \[[E:F]_s\leqslant [E:F]\] with equality if and only if $E/F$ is separable. Combining these facts, \[|\hbox{Aut}(E/F)|\leqslant [E:F]\] with equality if and only if $E/F$ is both normal and separable. Corollary 6 on page 562 is also equivalent to this fact.

Note 3: Definition of a splitting field

Dummit and Foote define normal extensions, but try to avoid using the term. However "normal" is a common term so I will use it. It is more or less equivalent to the term splitting field.

Dummit and Foote (on page 536) define the splitting field of a polynomial, but then in the definition of a normal extension (on page 537) they talk about the splitting field of a collection of polynomials. Indeed, if the collection is a finite set, $f_1, \cdots, f_n$, then it is obvious from the definition that the splitting field of this collection of polynomials is the same as the splitting field of the single polynomial $f_1 \cdots f_n$. The more general language of the definition of a normal extension is only needed for extension that are algebraic, but infinite, such as $\mathbb{Q} \left( \sqrt{2}, \sqrt{3}, \sqrt{5}, \cdots \right)$, which is a splitting field for an infinite collection of polynomials. Since we can't multiply an infinite number of polynomials together, the more general notion is needed. In problem 13.4 from HW2, the field extension is assumed to be finite, so we will interpret splitting field to mean the splitting field of a polynomial.

Note 2: Eisenstein's Criterion and Gauss' Lemma

In addition to our readings in Chapter 13, you should be familiar with Sections 3 and 4 of Chapter 9. In particular, you should understand Eisenstein's criterion (Proposition 13 on page 309) and Gauss's Lemma (Proposition 5 on page 303). For example consider how to prove that the polynomial $x^4-5$ is irreducible over $\mathbb{Q}$. By Eisenstein's criterion, it is irreducible in $\mathbb{Z}[x]$, and then by Gauss' Lemma, it is irreducible in $\mathbb{Q}[x]$.

Note 1: The bracket notation

Let $F$ be a field and $E$ a field containing $F$. If $\alpha\in E$ we will denote by $F(\alpha)$ the smallest field containing $F$ and $\alpha$. Occasionally we will use another notation: $F[\alpha]$ denotes the smallest ring containing both $F$ and $\alpha$. Equivalently it is the smallest algebra over $F$ containing $\alpha$. Sometimes $F[\alpha]$ and $F(\alpha)$ are the same, sometimes they are different.

Dummit and Foote use the notation $F(\alpha)$ but not the notation $F[\alpha]$. For them, the square brackets always indicate a polynomial ring. But the notation $F[\alpha]$ is fairly standard, and very useful. It is used in Lang's Algebra, for example. An objection to this notation is that with $x$ an indeterminate, the notation $F[x]$ is used to denote the polynomial ring in one variables over $F$. This is not actually inconsistent, however, since $F[x]$ is a subring of a field $E$, the field of fractions of $F[x]$ (also known as the field of rational functions). And with $x\in E$, the two things meant by $F[x]$ are the same!