Math 418, Intro to Abstract Algebra II

Spring 2022

Course Description

This is a second course in abstract algebra, covering the following topics:

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 24, the campus bookstore still has the wrong text listed for this course, so you will need to purchase this elsewhere.

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:

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.