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. 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
e-book collection. Another nice book is, which is also freely
available online is:
As per university policy, our first week will be online. The lectures will be held live rather than
be pre-recorded. 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.
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
1:30-4:30am in our usual classroom.
- Jan 19
- Jan 21
- 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
- Feb 14
Limitations of straightedge and
- Feb 16
Takehome #1 due;
- Feb 18
- Feb 21
Algebraically closed fields; the
Fundamental Theorem of Algebra.
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
- 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
- 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
- Mar 25
The Fundamental Theorem of Galois Theory II.
HW 6 due.
- Mar 28
Possible Galois groups and the
- Mar 30
Galois groups of polynomials.
- Apr 1
Solving equations by radicals; solvable
- Apr 4
Characterizing solvability by
- Apr 6
Introduction to Algebraic Geometry.
HW 7 due.
- Apr 8
Radical ideals and the
- 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
Takehome #2 due.
- Apr 22
Topology of curves and function fields
- Apr 25
Rational functions and field extensions.
- Apr 27
- Rational functions and field extensions II.
HW 9 due.
- Apr 29
- May 2
Cayley graphs and branch covers.
- May 4
Branched covers and the Riemann Existence Theorem.
HW 10 due.