M230C SPRING 2019 FINAL EXAM INSTRUCTIONS

EXAM LOGISTICS: Tuesday, June 11 2019, 1:30-3:30pm in RH440R.

1. There will be 5 problems: 2 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 Sections 14.2 -- 14.7 and 11.4, 12.2, 12.3.

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) Fundamental theorem of Galois theory.
(b) Determining Galois groups in simple situations.
(c) Finite fields.
(d) Composite extensions involving Galois extensions and calculations of degrees/Galois groups for composites.
(e) Simple extensions and criteria of simplicity. Determining simple generators.
(f) Determining minimal polynomial of a given element over a given field.
(g) Determining Galois groups of a given polynomial over Q.
(h) Cyclotomic and abelian extensions.
(i) Radical and solvable extensions.
(j) Determinants
(k) Rational canonical form of a matrix/opertor and its applications.
(k) Jordan canonical form of a matrix/opertor and its applications.

Of course, knowledge of background material will be necessary for aporoaching the problems.

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 examm.

M230

Last Modified: June 6, 2019