Math 418, Intro to Abstract Algebra II
This is a second course in abstract algebra, covering the
- 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
- 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
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
Reid, Undergraduate Algebraic Geometry.
Your course grade will be based on:
You can view your HW and exam scores here.
- 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.
- Jan 21
- Jan 23
- 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
- Feb 16
Limitations of straightedge and
- Feb 18
Takehome #1 due; Solutions.
- Feb 20
- Feb 23
Algebraically closed fields; the
Fundamental Theorem of Algebra.
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
- 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
- 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
- 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
- Apr 1
Galois groups of polynomials.
- Apr 3
Solving equations by radicals; solvable
- Apr 6
Characterizing solvability by
- Apr 8
Introduction to Algebraic Geometry.
HW 7 due.
- Apr 10
Radical ideals and the
- 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
Takehome #2 due.
- Apr 24
Topology of curves and function fields
- Apr 27
Rational functions and field extensions
- Apr 29
- Rational functions and field extensions II.
HW 9 due.
- May 1
- May 4
Cayley graphs and branch covers.
- May 6
Branched covers and the Riemann Existence Theorem.
HW 10 due.