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.

- Lang's
*Algebra*Third Edition, - Galois Theory by Emil Artin.

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

- My office is 383U on the third floor of the math building.
email:
bump@math.stanford.edu . Phone: 650-723-4011. - There will one in-class midterm (perhaps February 9) and one project, which will serve as a take home midterm. This will be assigned February 15 and you will have two weeks to do it. There will also be a final exam.
- Homeworks will be due on Wednesdays. Note that this is not a class day. You may give me the homework in person, email me a pdf file if you use latex, or put it in my mailbox. Homeworks will not be accepted after solutions are posted so try to get them done on time.

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.

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.

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

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

- Use the email address bump at math dot stanford dot edu.
- Please put Math 121 in the subject heading.

Homework | Dummit and Foote | Solutions | |||||

HW1 (due January 18) |
| HW1 Solutions | |||||

HW2 (due January 25) |
| HW2 Solutions | |||||

HW3 (due February 1) |
| HW3 Solutions | |||||

HW4 (due February 8) |
| HW4 Solutions | |||||

HW5 (due February 15) |
| HW5 Solutions | |||||

HW6 (due February 22) |
| HW6 Solutions | |||||

March 1: Project due | (March 8 will be last HW) | ||||||

HW7 (due March 8) |
| HW7 Solutions |

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.

- Scipione del Ferro (1465-1526).
- Niccolo Fontana Tartaglia (1499-1577).
- Gerolamo Cardano (1501-1567).
- Ludovico Ferrari (1522-1565).
- Rafael Bombelli (1526-1572).

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.

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.

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 $\mathbb{Q}[x]$.

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!