Math 418, Intro to Abstract Algebra II

Spring 2015

Course Description

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


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.


Your course grade will be based on: You can view your HW and exam scores here.


Jan 21
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.

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