Math 114 -- Second Course in Abstract Algebra (UC Berkeley)

Basic Information

  • Faculty: Adam Topaz
  • Lecture: TuTh 11:00am–12:30pm at B1 Hearst Annex
  • Office Hours: Tuesdays and Thursdays 1:30pm to 2:30pm. Wednesdays 11:00am to 12:00pm.
  • Office Number: 889 Evans.
  • Syllabus: Click here for the full syllabus.

Important Dates/Deadlines

Literature

  • Galois Theory by J.-P. Escofier
  • … other sources will be discussed in class, as needed.

Note: A personal electronic copy of the text is freely available to Berkeley students from SpringerLink. In order to access the free copy of the text on SpringerLink, you must be connected to the Berkeley campus network (e.g. airbears). You could also use Berkeley’s proxy server – here’s a link with more info.

List of Topics

The primary focus of this course is Galois Theory, including all of the basic theory leading up to the Fundamental Theorem of Galois Theory and the Galois Correspondence. We will begin the course with a very brief review of some of the required material from Math 113 (First Course in Abstract Algebra). As for the bulk of the class, I expect to cover most of the material from Chapters 1-12 of the text. If time permits, we will also discuss some of the material from Chapters 14 and 15, and possibly some other topics in abstract algebra.

Here is a tentative list of topics:

  • Some history and motivation on solving polynomial equations.
  • Symmetric polynomials, elementary symmetric polynomials, resultants and discriminants of polynomials.
  • Field extensions, towers of fields, algebraic elements, algebraic extensions, transcendental elements, adjoining roots.
  • Ruler and compass constructions, constructible numbers.
  • Embeddings of fields, extensions of isomorphisms, normal extensions, splitting fields.
  • The Galois group, the Galois correspondence, the Fundamental Theorem of Galois Theory.
  • Roots of unity and cyclotomic extensions, cyclic extensions.
  • Solvable extensions, solvability by radicals.

If there’s time:

  • Finite fields and their Galois theory.
  • Separable extensions, Galois theory over non-perfect fields.

Grading

Final grades will be based on homework, one midterm, and a final exam. Your final grade will be roughly calculated using the following formula:

  • Homework: 30%
  • Midterm: 30%
  • Final: 40%

Important Note: There will be no make-up exams. If you have a legitimate conflict with the scheduled date/time for the midterm (which will be established before the first lecture), you should let me know within the first two weeks of class.

Homework

Homework will be assigned on a weekly basis (with the exception of the week of the midterm). Homework assignments will be posted below. Late homework will not be accepted under any circumstances, but your lowest two homework grades will be dropped.

Some Review Material

Here are some practice problems from the Math 113 class I taught in the Fall 2013 semester.

Note: Most of these problems are of the same style: Find an example or prove that none exists. Even in cases where the directions ask for an example (e.g. an element in a ring), the problems should still be treated as if “no example” is a possibility, and in such cases you should provide a proof that no example exists.