Mathematics 2F, Solutions to exercises, 20162017
II. Number theory
1. (a) If a | b and b | c, then b = ua and c = vb with u, v Z, hence
c = vb = v(ua) = (vu)a
with vu Z, hence a | c.
(b) Suppose that a | b and b | a. Then there are integers r, s such that
Solutions and Comments
Q1
1 2
Find K R so that the implication
x +5
| x + 2| 1 =
+ 1 K | x + 2|
x1
is true for all x R. As always, make sure you fully justify your answer.
This exercise is very similar to example 1.8 from lectures. The first
5
step is t
Solutions and Comments
1 2
.
If youve not seriously tried these exercises, please dont look at these solutions
and comments, until you have. Youll
get the most benefit from reading these
comments, when youve first thought
hard about them yourself, even if
Mathematics 2F, Solutions to exercises, 20162017
IV. Isometries of the plane.
1. (a) Let Q be the projection of P (2, 3) on L. Then Q has coordinates
(2, 3) + t(1, 2) = (2 + t, 3 + 2t).
We have
2 + t + 2(3 + 2t) = 2,
hence t =
56 .
The reflection R of P
Mathematics 2F, Solutions to exercises, 20162017
I. Sets, functions, cardinality and equivalence relations
1. Suppose firstly that x
/ A B. Then it is not true that x belongs to both A and B. Then x
does not belong to at least one of these sets A, B. Hen
Ex Sheet
9
2A Multivariable Calculus 2013
Tutorial Exercises
T1
Evaluate
P
xy2 dx + x4 y dy,
where P is the arc of the parabola y = 2x2 from A(0, 0) to B(1, 2). (a)
by parametrising the curve, (b) using x, y coordinates.
Solution
(a) Parameterise P
y = 2t
Chapter 0
Revision of dierentiation, integration
and vector algebra from level 1
0.1
Revision of dierentiation
(Stewart (Ed. 7): Chapter 2, p103.)
0.1.1
Three important rules for dierentiation
Product Rule This rule is for dierentiating the product of fun
2. Number theory
2.1.
Greatest common divisors and the Euclidean algorithm.
Number theory is the study of the integers, especially the positive integers. Positive
integers are called natural numbers. The set consisting of positive integers is
denoted N, a
Definition 3.21. Let g be an element in a group G. The the subgroup of G
generated by g, denoted hgi, is the subgroup given by
hgi= cfw_g n : n Z .
The order of g, denoted |g|, is the order of the subgroup hgi.
Example 3.22. The subgroup of Z generated by
1. Polynomial equations
In this introductory section we shall discuss the historical background of Galois
theory. More precisely, we shall have a look at the problem of solving polynomial
equations. The importance of such equations was recognised very ear
34
CHRISTIAN VOIGT
5. Automorphisms, normality and separability
Let F jK be a _eld extension. We write Aut(F jK) for the set of all _eld automorphisms _ : F ! F which _x K, that is, for which we have _() = for all
2 K _ F . The set Aut(F jK) is a group wi
_
a b
_
_
_
7. The quintic equation
In this section we show that a quintic polynomial equation is in general not solvable by radicals. This result was one of the original motivations for the
development of Galois theory. Its proof relies on the Galois cor
40
CHRISTIAN VOIGT
6. Galois theory
This section contains the core results of Galois theory. They provide a strong
link between sub_elds of certain _eld extensions F jK and subgroups of the corresponding automorphism group Aut(F jK).
The _eld extensions w
a2
2
q
a2 q
p
_
_
3 2a3
a2 a2 a3 p
q
q a2
p
q
q
p
1a
2
2
2
+
a2q u =q2 2 p3 q
q a2 q
q
pp 32; q q27 p p3 a2 p
q a
r
r
_
_
_
a2
p
1.3. Cubic equations. This is the case n = 3. The history of the cubic equation is quite
interesting, but we shall not review
n
1
1
1
1
i
1
i
17
3. Rings of polynomials
In this section we take a closer look at rings of polynomials.
We shall study
irreducible polynomials and methods for determining irreducibility.
3.1. Polynomials. We begin by recalling the general de_nition of a
2. Rings, fields and integral domains
In this section we discuss some basic material on rings, integral domains and
_elds. For further information and details we refer to Fraleigh's book [2].
2.1. Rings. Let us begin with the de_nition of a ring.
De_nitio
4H TOPICS IN ALGEBRA MOCK EXAM
This paper contains ve questions. The rst two are compulsory and
contribute 20 = 2 10 marks. From the last three, candidates must
choose two that contribute the remaining 30 = 2 15 marks.
The bold face numbers such as (3) in
Some day, The n th Some month, Some year
Some time to Another time
EXAMINATION FOR THE DEGREES OF
M.A., B.Sc. AND M.Sci.
4H HONOURS MATHEMATICS
4H Topics in Algebra
An electronic calculator may be used provided that it does not have
a facility for either
4H TOPICS IN ALGEBRA MOCK EXAM
This paper contains ve questions. The rst two are compulsory and
contribute 20 = 2 10 marks. From the last three, candidates must
choose two that contribute the remaining 30 = 2 15 marks.
The bold face numbers such as (3) in
p
p
p
p
Field extensions
A central concept in Galois theory is the notion of a _eld extension. In this
section we collect important de_nitions and results related to _eld extensions, and
discuss some examples. We will also study geometric applications con
Galois Theory
Sheet 0
Solutions
1.
Write down the denition of a ring and of a ring homomorphism.
See lecture notes.
2.
Remind yourself what it means for a ring to be commutative. Give two examples of rings,
one commutative, and one non-commutative.
Exampl
Galois Theory
Sheet 3
Field extensions
The solutions to starred questions are to be handed in on Monday 3rd March at 2 pm.
Please put your homework in the folder on my oce door (330).
1.* Write down the denition of a vector space over a eld K. Verify that
Galois Theory
Sheet 2
Solutions
1.
Let K = F3 . Find quotient and remainder when performing long division of f (x) =
2x3 + 2x2 + x + 1 by g(x) = 2x2 + 2 in K[x].
We obtain q(x) = x + 1, r(x) = 2x + 2.
2.* Consider the polynomials f (x), g(x) F5 [x] given
Galois Theory
Sheet 2
Polynomial rings
The solutions to starred questions are to be handed in on Monday 17th February at 2
pm. Please put your homework in the folder on my oce door (330).
1.
Let K = F3 . Find quotient and remainder when performing long di
Galois Theory
Sheet 1
Solutions
1.
Let R be a ring and let u R be a unit. Show that the element v R satisfying
vu = 1 = uv is uniquely determined.
Let w R be another element such that wu = 1 = uw. Then
w = 1w = (vu)w = v(uw) = v1 = v.
We conclude that the
Galois Theory
Sheet 1
Rings, elds, and integral domains
The solutions to starred questions are to be handed in on Monday 3rd February at 2
pm. One of these questions will be marked and the homeworks will be returned in tutorial.
Please put your homework i
Galois Theory
Sheet 3
Solutions
1.* Write down the denition of a vector space over a eld K. Verify that if F |K is a eld
extension then F is naturally a K-vector space.
A K-vector space consists of an abelian group V under addition together with a map
K V
Specimen Paper
Duration: 90 min
EXAMINATION FOR THE DEGREES OF
M.A., B.Sc. AND M.Sci.
4H HONOURS MATHEMATICS
Galois Theory
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphic
Galois Theory
Sheet 0
Some things you should already know
The purpose of this sheet is to check your knowledge of material from previous semesters.
You should be able to answer most of these questions rather quickly. We will revise some of
this material d