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 course.
3.1. Normalizers
We need one more tool before we ca
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 have to do.
No calculators or notes will be allowed.
You
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 is called the conjugacy class of x.
This is also the se
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 have to do.
No calculators or notes will be allowed.
You
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 of the subgroups of G, and state which
are normal. Find
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 Galois group. We develop these results in
the rst sectio
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 root.
Find the exact roots of f = x3 15x 4 by factoring.
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 = 1 G there exists an
x X such that gx = x. That is, onl
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 there a Q-automorphism of L
with (i 3) = 1 + i 3? Expla