# MA3202 Galois theory – spring 2015

## Lecturer

Karin Erdmann
Room 834, Sentralbygg II.
Telephone 73 59 xx xx.
E-mail: erdmann at maths.ox.ac.uk
Office hours Wednesdays 11-12

no office hour Wednesday 6 May
(there will be office hours the following weeks)

## Lectures and exam

Tuesday 08.15-10.00, M24 (note change)
Friday 08.15-10.00, M24
First lecture is Tuesday January 6
Exam: June 1 at 9.00-13.00

Reference group meetings: Thursday 5 Feb
Thursday 19 March (10am)

Easter break
We start again Tuesday 14th April
(ie no lecture on Friday 10th)

## Lecture Log

 Week 1   first half of ch 11
Week 2   second half of ch11. Started ch 15
Week 3   ch.15 up to 3.2
Week 4   ch.15 up to 4.2
Week 5   ch.16 Sections 1 and 2
Week 6   ch.16 Section 5. From section 4: 4.6, 4.1, 4.3, 4.4, 5.1(b).
Week 7   ch.17 up to 1.6
Week 8   ch.17 examples and 2.1
Week 9   ch.17 finite fields. Start on ch18.1
Week 10  ch.18 Sections 1 and 2
Week 11  ch.18 Start on Section 3
Week 12  ch.18 finishing Section 3
Week 13  ch.18 Section 5

## Textbook

P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul
Basic Abstract Algebra
Cambridge University Press Second Edition ISBN: 0-521-46629-6

## Syllabus

Contained in the chapters 11, 15, 16, 17, 18 in the textbook

## Exercises

Going over solutions: Fridays during the first hour

#### Exercise 1

Chapter 11.1: questions 1, 2, 3, 8
Chapter 11.3: questions 2, 4, 8
Problem (posed by Fermat)
Prove that the Diophantine equation y^2+2=x^3 has only the integer solutions y=5 or -5 and x=3.
[Hint: Use that Z[\sqrt{-2}] is a Euclidean domain with norm N(a+b\sqrt{-2}) = a^2+2b^2.]

#### Exercise 2

Chapter 15.1: 2, 4, 6(b)
Chapter 15.2: 2, 4
Chapter 15.3: 2, 4, 6, 10
Problem 4 (Continuation Exam August 2013)

#### Exercise 3 (different from previous year)

Construct splitting fields over Q for the polynomials x^3-1, x^4+5x^2+6, x^6-8.
Find the degrees of these fields over Q.
Chapter 16.1: 3, 4, 8
Chapter 16.2: 2, 4
Exam May 2013 problems 2b), 3, 7.

#### Exercise 4

Do True-or-False exercise: true-or-false.pdf
Continuation Exam August 2013: Problem 5
Exam, May 2013: Problems 1, 2a, 6.

#### Exercise 5

Exam May 2009 Problem 3
Cont Exam Aug 2013 Problem 2
Exam May 2013 Problem 5
Exam May 2012 Problem 5
Exam May 2011 Problem 2
Exam May 2010 Problems 1, 5
Exam May 2009 Problem 2.

#### Exercise 6

Cont Exam Aug 2013 Problem 3
Exam 2012 Problems 1, 4
Exam 2011 Problems 1, 3, 4
Exam 2009 Problem 5
Exam 2014 Problems 4, 5, 6.

#### Exercise 7

Exam May 2010 Problems 3,4,6
Exam 2006 Problems 1, 5
Exam 2004 Problem 5
Rest of Exam 2014

## Old exams

#### Some lecture notes

On Ch 15 ch15variation.pdf
Ch 16, Splitting fields ch16-splittingfields.pdf
Ch. 16, Separable extensions ch16-_separable.pdf