THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
November 2007
MATH5725
GALOIS THEORY
(1) TIME ALLOWED 3 HOURS
(2) TOTAL NUMBER OF QUESTIONS 6
(3) ATTEMPT ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) THIS PAPER MAY BE R
School of Mathematics and Statistics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Assignment 1. Due thursday week 5 1
Below, F will always denote a eld.
1. What is [Q( 2, 5 2) : Q]? Write down a Q-basis for Q( 2, 5 2).
2. Let K be t
FACULTY OF SCIENCE
SCHOOL OF MATHEMATICS AND
STATISTICS
MATH5725
GALOIS THEORY
Session 2, 2011
Course Outline
Lecturer/Tutor: Daniel Chan
E-Mail: danielc@unsw.edu.au
Webpage: web.maths.unsw.edu.au/danielch
Oce: Red Centre (East Wing) Room 4104
Oce Phone N
School of Mathematics and Statistics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Assignment 2. Due tuesday week 12 1
1. Let 2 be the eld with two elements. Compute the following: i)
F
[F2 ( 3 t) : F2 (t)], ii) [F2 ( 3 t) : F2 (t)]s and
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
November 2009
MATH5725
GALOIS THEORY
(1) TIME ALLOWED 2 HOURS
(2) TOTAL NUMBER OF QUESTIONS 4
(3) ATTEMPT ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) THIS PAPER MAY BE R
School of Mathematics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Problem Set 2: Splitting elds 1
Below, F will always denote a eld.
1. Write down a splitting eld for x3 1 over Q in the form i) Q() for
some element C and ii) Q[x]/ p(x)
School of Mathematics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Problem Set 4: Galois extensions & correspondence
1
1. Which of the following eld extensions is (nite) Galois? i) Q( 3 7)/Q
2i/3
2i/3
iii)
)/Q iv) Q( 3 7, e )/Q(e2i/3
These notes were taken down by me (Boris Lerner) during the Galois
Theory course taught by Dr Daniel Chan in the second session of 2007 at
UNSW. I made a few brief remarks of my own here and there but in general this is my best attempt to reproduce exactl
School of Mathematics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Problem Set 6: Solvability by Radicals.
Finite & Cyclotomic Fields 1
1. Find the Galois groups of the following polynomials over Q: i) x2 3
ii) x4 2 iii) x3 2 iv) x5 4x
School of Mathematics and Statistics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Problem Set 5: Radical extensions & Solvability
1
1. Let K/F be a Galois extension with Galois group G. For any K,
show that f (x) := G (x () F [x] so the
School of Mathematics & Statistics
University of New South Wales
MATH5725: Galois Theory (2011,S2)
Problem Set 3: Constructing Field Automorphisms.
Normal & Separable Extensions 1
1. Let be a primitive 5-th root of unity. Show that F = Q( 5 3, )
is a spli