Math 418, Intro to Abstract Algebra II
Spring 2015
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.
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.
Grading
Your course grade will be based on:
 Weekly homework assignments: (20%) These will typically be
due in class 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 onehour exam will be held in
our usual classroom, on Monday, March 9.
 Final exam: (35%) This will be Tuesday, May 12 from
811am in our usual classroom.
You can view your HW and exam scores here.
Schedule
 Jan 21

Introduction.
 Jan 23

Euclidean Domains.
 Jan 26

Principal Ideal Domains.
 Jan 28

PIDs are UFDs.
HW 1 due.
 Jan 30

Which polynomial rings are UFDs?
 Feb 2

R[x] is a UFD
if R is; irreducibility criteria.
 Feb 4

Field extensions I.
HW 2 due.
 Feb 6

Field extensions II.
 Feb 9

Algebraic numbers and extensions.
 Feb 11

More on algebraic extensions.
HW 3 due.
 Feb 13

Field multiplication as linear
transformations.
 Feb 16

Limitations of straightedge and
compass.
 Feb 18

Constructable numbers.
Takehome #1 due; Solutions.
 Feb 20

Splitting fields.
 Feb 23

Algebraically closed fields; the
Fundamental Theorem of Algebra.
Here are
various proofs of the of the
Fundamental Theorem of Algebra.
 Feb 25

Polynomials with distinct roots; separability criterion.
HW 4 due.
 Feb 27

Finite fields; cyclotomic fields.
 Mar 2

Cyclotomic polynomials and
applications.
 Mar 4

Introduction to Galois Theory.
HW 5 due.
 Mar 6

Galois groups of splitting fields.
 Mar 9

In class midterm.
 Mar 11

Primitive extensions and
minimal polynomials.
 Mar 13

No class. Read about Fundamental Theorem of Algebra instead.
 Mar 16

Finite fields and degrees of fixed fields.
 Mar 18

The Fundamental Theorem of Galois
Theory I.
 Mar 20

The Fundamental Theorem of Galois Theory II.
HW 6 due.
 Mar 21
 Spring Break starts.
 Mar 29
 Spring Break ends.
 Mar 30

Possible Galois groups and the
discriminant
 Apr 1

Galois groups of polynomials.
 Apr 3

Solving equations by radicals; solvable
groups.
 Apr 6

Characterizing solvability by
radicals.
 Apr 8

Introduction to Algebraic Geometry.
HW 7 due.
 Apr 10

Radical ideals and the
Nullstellensatz.
 Apr 13

Decomposition into irreducibles and
more on Hilbert's results. Also, here is a
proof of the Nullstellensatz for arbitrary fields.
 Apr 15

Functions on varieties.
HW 8 due.
 Apr 17

Projective space I.
 Apr 20

Projective space II.
 Apr 22

Elliptic curves.
Takehome #2 due.
 Apr 24

Topology of curves and function fields
of varieties.
 Apr 27

Rational functions and field extensions
I.

 Apr 29
 Rational functions and field extensions II.
HW 9 due.
 May 1

Branched covers.
 May 4

Cayley graphs and branch covers.
 May 6

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