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
e-book 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 one-hour exam will be held in
our usual classroom, on Monday, March 9.
- Final exam: (35%) This will be Tuesday, May 12 from
8-11am 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.