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
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 simil
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, whe
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
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
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) usi
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
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.
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 sub
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. T
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
_
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
developme
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 correspond
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
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. Polyno
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. Rin
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 remainin
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 use
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 remainin
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
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
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 t
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.
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 = F
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. 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 w
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
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 facilit
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