MWF 12pm, Lunt 102 Instructor: Florian Herzig Office: Lunt 309 Phone: 467-1898 E-mail: my last name at math dot northwestern dot edu Office Hours: Wed 5:00-6:30, Fri 1:30-3:00

Note: This page will be updated as the course progresses.

Course description

This is the first quarter of the graduate algebra sequence. This term the main focus is on some group theory and on Galois theory.

Topics to be covered

Group theory: topics including solvable groups, free groups, and amalgamated product;
interlude on universal properties, functors, natural transformations

Galois theory: normal and separable extensions, fundamental theorem of Galois theory, algebraic closure, solvability of polynomial equations

Infinite Galois theory, profinite groups, and p-adic numbers

Prerequisites

I will assume that you have taken a standard honors algebra course on groups, rings, and fields.

Textbook

Serge Lang: Algebra, revised 3rd ed., Springer or Addison-Wesley

We will not follow Lang very closely.
Other general books on algebra:

Nathan Jacobson: Basic Algebra (2 volumes)

On group theory:

Joseph Rotman: An introduction to the theory of groups (GTM)

Derek Robinson: A Course in the Theory of Groups (GTM)

Jonathan Alperin, Rowen Bell: Groups and Representations (GTM); beautifully written, but less comprehensive

Krull's original paper on infinite Galois extensions,
Math. Ann. 100, 687-698 (1928);
Baer's review of Krull's paper in Zentralblatt
I found this very interesting. For example, it seems "separable" was called "algebraic of first kind". Also keep
in mind that Hausdorff's definition of a topological space, which included the Hausdorff axiom, wasn't much more than 10 years old at that time.
It seems Krull was
initially mainly interested in the case of countable infinite extensions (and in that case the Galois group is metrisable); he thanks
John v. Neumann for the observation that this should generalise (see footnote 11).
Note that the "finite cover" notion of compactness wasn't introduced until 1929, which explains why Krull only shows that Galois groups are compact in the countable case.
Finally he discusses the case of
an "ideal-cyclic" (i.e., procyclic) Galois group.

Krull mentions that Dedekind (1901) was the first to realise that intermediate fields and subgroups of the Galois group aren't in natural
bijection for infinite extensions. Dedekind's example was the extension of the rationals obtained by adjoining p^{n}-th roots of unity for all n,
for a fixed prime p. (The review of Dedekind's paper at Zentralblatt is almost incomprehensible in its old-fashioned mathematical language...)