Math 418, Intro to Abstract Algebra II
Spring 2022
Course Description
This is a second course in abstract algebra, covering the
following topics:
 Rings: Polynomial rings, fields of fractions, and
other examples. Euclidean domains, principal ideal domains, and
unique factorization domains.
 Fields: Field extensions and Galois Theory.
Solvability of equations by radicals. Ruler and compass
constructions.
 Algebraic geometry: Basic correspondence between ideals
and varieties in affine and projective space, with examples such as
elliptic curves. Decomposition into irreducibles, Hilbert's
Nullstellensatz, and connections to Galois Theory.
Prerequisites:
The needed background for this course is Math 417, Intro to
Abstract Algebra. Math 427 is also fine, though there is some overlap
between that course and this one.
Required text: Dummit and Foote, Abstract Algebra,
3rd Edition, 944 pages, Wiley 2003. As of Jan
5, the campus bookstore has a different text listed for this course,
but this should be corrected shortly.
Supplementary texts: For the final part of the course
covering algebraic geometry, one good reference beyond Chapter 15
of Dummit and Foote is:
Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms,
Springer Undergraduate Texts in Mathematics.
You can get it in PDF format via the library's
ebook collection. Another nice book is, which is also freely
available online is:
Reid, Undergraduate
Algebraic Geometry.
First week
As per university policy, our first week will be online. The lectures will be held live rather than
be prerecorded. I will email everyone enrolled in the class the
Zoom link on Tuesday, January 18, in time for my office hour; if you
have not yet registered or are auditing, send me an email.
Grading
Your course grade will be based on:
 Weekly homework assignments: (20%) These will typically be
due on Wednesday. Late homework will not be accepted; however, your
lowest two homework grades will be dropped, so you are effectively
allowed two infinitely late assignments. Collaboration on homework is
permitted, nay encouraged. However, you must write up your solutions
individually and understand them completely.
 Two takehome midterms: (12.5% each) These are glorified HW
assignments that you are to work on individually. They will replace
the usual HW for two weeks of the term.
 In class midterm: (20%) This 50 minute exam will be held in
our usual classroom, on Monday, March 9.
 Final exam: (35%) This will be Wednesday, May 11 from
1:304:30am in our usual classroom.
Schedule
 Jan 19

Introduction.
 Jan 21

Euclidean Domains.
 Jan 24

Principal Ideal Domains.
 Jan 26

PIDs are UFDs.
HW 1 due.
 Jan 28

Which polynomial rings are UFDs?
 Jan 31

R[x] is a UFD
if R is; irreducibility criteria.
 Feb 2

Field extensions I.
HW 2 due.
 Feb 4

Field extensions II.
 Feb 7

Algebraic numbers and extensions.
 Feb 9

More on algebraic extensions.
HW 3 due.
 Feb 11

Field multiplication as linear
transformations.
 Feb 14

Limitations of straightedge and
compass.
 Feb 16

Constructible numbers.
Takehome #1 due;
 Feb 18

Splitting fields.
 Feb 21

Algebraically closed fields; the
Fundamental Theorem of Algebra.
Here are
various proofs of the of the
Fundamental Theorem of Algebra.
 Feb 23

Polynomials with distinct roots; separability criterion.
HW 4 due.
 Feb 25

Finite fields; cyclotomic fields.
 Feb 28

Cyclotomic polynomials and
applications.
 Mar 2

Introduction to Galois Theory.
HW 5 due.
 Mar 4

Galois groups of splitting fields.
 Mar 7

In class midterm.
 Mar 9

Primitive extensions and
minimal polynomials.
 Mar 11

No class, read about The
Fundamental Theorem of Algebra instead.
 Mar 12
 Spring Break starts.
 Mar 20
 Spring Break ends.
 Mar 21

Finite fields and degrees of fixed fields.
 Mar 23

The Fundamental Theorem of Galois
Theory I.
 Mar 25

The Fundamental Theorem of Galois Theory II.
HW 6 due.
 Mar 28

Possible Galois groups and the
discriminant
 Mar 30

Galois groups of polynomials.
 Apr 1

Solving equations by radicals; solvable
groups.
 Apr 4

Characterizing solvability by
radicals.
 Apr 6

Introduction to Algebraic Geometry.
HW 7 due.
 Apr 8

Radical ideals and the
Nullstellensatz.
 Apr 11

Decomposition into irreducibles and
more on Hilbert's results. Also, here is a
proof of the Nullstellensatz for arbitrary fields.
 Apr 13

Functions on varieties.
HW 8 due.
 Apr 15

Projective space I.
 Apr 18

Projective space II.
 Apr 20

Elliptic curves.
Takehome #2 due.
 Apr 22

Topology of curves and function fields
of varieties.
 Apr 25

Rational functions and field extensions.

 Apr 27
 Rational functions and field extensions II.
HW 9 due.
 Apr 29

Branched covers.
 May 2

Cayley graphs and branch covers.
 May 4

Branched covers and the Riemann Existence Theorem.
HW 10 due.