CHAPTER 3
Further Group Theory
In this chapter we prove Sylows Theorem, and also dene nilpotent groups. If Sylows Theorem is
accepted on faith, then this chapter is not needed for the rest of the cour
4301 Practice Exam
2011
The exam will take place on Friday 11th November from 8:00 to 10:00 am. (2 hours, no reading time).
There will be 5 questions. There may be some choice of which questions you h
CHAPTER 2
Groups
2.1. Review of Groups
Let G be a group. If g G then the conjugate of x by g is the element y = g 1 xg . We say that x
and y are conjugates. The set of all g 1 xg as g varies across G
4301 Practice Exam
2011
The exam will take place on Friday 11th November from 8:00 to 10:00 am. (2 hours, no reading time).
There will be 5 questions. There may be some choice of which questions you h
4301 Assignment 4
Question 1.
Due Thursday 27th October at 4 pm
[Fundamental Theorem Example] Let f = x3 2. Let K be the splitting eld of f
over Q. Calculate the Galois group G = G al(K/Q). Find all o
CHAPTER 8
Solving Polynomial Equations II
Solving a polynomial exactly involves working in extensions like K ( n a ). Thus we need some facts
about extensions L/K such that L/K is Galois with cyclic G
4301 Assignment 1
Due Thursday 18th August, 4 pm
Question 1. [Resultants; Simplifying results in Cardanos method]
(a)
(b)
(c)
(d)
Find a non-zero polynomial with integer coefcients with 3 + 3 3 as a r
4301 Assignment 2
Due Thursday 1st September at 4 pm
Answer Questions 13 and one of 46.
Question 1.
[Faithful actions]
Let G be a group acting on a set X . The action is called faithful if for any g =
4301 Assignment 3
Question 1.
Due Thursday October 6th
[Isomorphism Extension] Show that i 3 and 1 + i 3 are roots of the polynomial
f = x4 2x3 + 7x2 6x + 12. Let L be a splitting eld of f over Q. Is