M230 SPRING 2019 MIDTERM INSTRUCTIONS

Exam logistics: Tuesday, May 14 2019, 4-5:20pm in RH440R instead of the lecture.

1. There will be 4 problems: 1 easy, 2 intermediate, and 1 more convoluted, although not too challenging.

2. The topics will cover the material in the book discussed in the lecture/discussions from Section 13.1, 13.2, 13.4, 13.5, 13.6, 14.1, 14.2 and 14.3.

3. There will be no question asking to reproduce a proof of a theorem from the book, instead, the problems will ask to use the theory we went through so far to
(a) solve some given problems about concrete field extensions.
(b) prove some simple general facts about field extensions.
The problems will be similar to homework problems and to recommended problems suggested on the homework assignment website.

4. Here are some topics to focus on:
(a) Basic knowledge about concrete fields and field extensions we went over.
(b) Basic constructions of field extensions as quotients corresponding to irreducible polynomials. Extending isomorphisms to such field extensions.
(c) Splitting fields. Extending isomorphisms to splitting fields.
(d) Separable polynomials and extensions.
(e) Finite fields, cyclotomic fields. Calculatins in F and in F[x] where F is a finite field. Cyclotomic polynomials.
(f) Galois extensions. Determining whether an extension is Galois.
(g) Calculating automorphism groups of field extensions, in particular calculating Galois groups of Galois extensions.

5. It will be important that you express yourself clearly and correctly. It will be expected that you write your arguments rigorously, but include just the right amount of information. Please take this into account when preparing for the midterm.

M230

Last Modified: May 13, 2019