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Introduction to

Algebraic Structures

Lecture Notes 2019

Janko Böhm, Magdaleen Marais

November 6, 2019

Contents

LIST OF FIGURES i

5.1 Cubic polynomials with zeroes − 1 and 2 and in- flection point at 0.................... 112 5.2 Half plane........................ 116 5.3 Parabola......................... 117

• 0 Introduction
• 1 Fundamental constructions
  • 1.1 Sets
  • 1.2 Mathematical induction
  • 1.3 Relations
  • 1.4 Maps
  • 1.5 Partial orderings and equivalence relations
  • 1.6 Exercises
• 2 Numbers
  • 2.1 The integers and division with remainder
  • 2.2 Fundamental theorem of arithmetic
  • 2.3 Greatest common divisor and Euclidean algorithm
  • 2.4 The Chinese remainder theorem
  • 2.5 Prime factorization
  • 2.6 Exercises
• 3 Groups
  • 3.1 Overview
  • 3.2 Groups and actions
    • 3.2.1 Basics
    • 3.2.2 Group actions
    • 3.2.3 Action by translation
  • 3.3 Normal subgroups
    • 3.3.1 Normal subgroups and the quotient group
    • 3.3.2 Homomorphism theorem
  • 3.4 Exercises
• CONTENTS
• 4 Rings and fields
  • 4.1 Basics
  • 4.2 The group of units of Z~n
  • 4.3 Ideals and quotient rings
  • 4.4 Integral domains and fields
  • 4.5 Exercises
• 5 Vector spaces
  • 5.1 Overview
  • 5.2 Gauß algorithm
  • 5.3 Vector spaces and bases
  • 5.4 Dimension
  • 5.5 Vector space homomorphisms
  • 5.6 Exercises
• 1 Knot List of Figures
• 2 Four points.
• 1.1 Complement
• 1.2 Union
• 1.3 Intersection
• 1.4 Graph of parabola.
• 1.5 Hyperbola
• 1.6 A non-injective map
• 1.7 A bijektive map and its inverse map.
• 1.8 Square root
• 1.9 Identity map R → R
• 1.10 Equivalence classes
• 1.11 Towers of Hanoi
• 1.12 How many shortest paths are there from A to B.
• 2.1 Two configurations of gearwheels.
• 3.1 The Platonic solids
• 3.2 Composition of two symmetries of the tetrahedron.
• 3.3 Rotational symmetry of the tetrahedron
• 3.4 Reflection symmetry of the tetrahedron
• 3.5 residue classes modulo
• 3.6 Exponential function
• 3.7 Example of a motion of R
• 3.8 Tetraedron
• 3.9 Regular 5 -gon
  • of the edges 3.10 Tetrahedron with diagonals connecting mid-points
• 3.11 Icosahedron with numbering of the vertices
• m ∉ M m is not an element of M
• m ∈ M m is an element of M
• N natural numbers
• N 0 natural numbers including
• Z integers
• Q rational numbers
• S with
• ⇒ from which it follows that
• ⇔ if and only if
• M ƒN complement of N in M
• M ∪ N union of N and M
• M ∩ N intersection of N and M
• ∀ for all
• ∃ there exists
• SM S number of elements of M
• M × N cartesian product of M and N
• 2 M power set of M
  • k= 1 ak sum n
• ∏nk= 1 ak product
• f (A) image of A under the map f
• f −^1 (B) preimage of B under the map f
• Graph(f ) graph of f
• f −^1 inverse of the bijective map f
• ∃ 1 there exists a unique
• ‰nkŽ binomial coefficient
• b S a b divides a
• a ≡ b mod m a congruent to b modulo m
• π (x) number or prime numbers less or equal to x
• ggT greatest common divisor
• lcm least common multiple LIST OF SYMBOLS iii
• S (X) group of self-mappings of X
• Sn symmetric group
• G 1 × G 2 cartesian product of G 1 and G
• Z~n group of residue classes
• Zn group of residue classes
• ker ϕ kernel of ϕ
• Im ϕ Image of ϕ
• ⟨E⟩ subgroup generated by E
• ord (g) order of g
• E (n) group of Euklidean motions
• Sym (M ) symmetry group
• Gm orbit of m under the action of G
  • G Stab (N ) stabilizer of the set N unter the action of
• Stab (m) stabilizer of m under the action of G
• [G ∶ H] index of the subgroup H ⊂ G
• deg (f ) degree of the polynomial f
• ϕ(n) Euler Phi-function, n ∈ N
• char (K) characteristic of K
• ϕΩ linear combination map with respect to Ω
• dim V dimension of V
  • Basen Ω und ∆ M ∆Ω(F ) Darstellende Matrix von F bezüglich der

LIST OF SYMBOLS iv

0. INTRODUCTION 2

of the equation xn^ + yn^ = zn.

Fermat’s last theorem was only proven 1995 (by A. Wiles) af- ter 350 years of work of many mathematicians, which involved introducing various new concepts in mathematics. Today, there are close connections of number theory to, for example, algebraic geometry, combinatorics, cryptography and coding theory. What is algebra? Algebra is a very diverse area of mathemat- ics, which discusses basic structures which are of key importance in all fields of mathematics, like groups rings and fields. That is, algebra studies the question, how one can introduce operations on sets, like the addition and multiplication of integer numbers. By combining methods from algebra and number theory, one can construct, for example, public key cryptosystems. Another connection of algebra and number theory arises from algebraic geometry, which studies solution sets of polynomial systems of equations in several variables^3. The simplest (but in practice the most important) special case are linear systems of equations over a field K (for example, K = Q, R, C the field of rational, real or complex numbers), the

(^3) For example, the common solution set of x (^2) + 2 y (^2) = 3 and 2 x (^2) +

y^2 = 3 , that is, the intersection of two ellipses, consists out of 4 points ( 1 , 1 ), (− 1 , 1 ), ( 1 , − 1 ), (− 1 , − 1 ), see Figure 2.

0

1

2

y

–1 (^0) x 1 2

Figure 2: Four points.

0. INTRODUCTION 3

core topic of linear algebra. Here, we solve systems

a 1 , 1 x 1 + ... + a 1 ,mxm = b 1 ⋮ an, 1 x 1 + ... + an,mxm = bn

with aij ∈ K, bi ∈ K for xj ∈ K (with i = 1 , ..., n and j = 1 , ...m). As an application of linear algebra, we will discuss the Google page rank algorithm. Let us also comment on an other special case, which however goes beyond the topics discussed in this course, that is, poly- nomial equations of higher degree in a single variable x. For example, one can ask for the solution set of the quadratic equa- tion ax^2 + bx + c = 0.

The solutions can be described, using radicals, as

x =

−b ±

b^2 − 4 ac 2 a

In a similar way, one can find expressions in terms of radicals for equations of degree d = 3 (Tartaglia 1535 , Cardano 1545 ) and d = 4 (Ferrari 1522 ), for d ≥ 5 the solutions can, in general, not be written in terms of radical any more. An important subtopic of algebra, the Galois theory, discusses when this is possible.

1. FUNDAMENTAL CONSTRUCTIONS 5

Example 1.1.3 The following sets of numbers are examples of sets: The numerical digits

{ 0 , 1 , 2 , ..., 9 } ,

the natural numbers

N = { 1 , 2 , 3 , ...} N 0 = { 0 , 1 , 2 , 3 , ...} ,

the integers Z = { 0 , 1 , − 1 , 2 , − 2 , ...} ,

the rational numbers

Q = ›

a b

S a, b ∈ Z, b ≠ 0.

The Symbol S is written in place of with.

Definition 1.1.4 If every element of the set N is also an ele- ment of the set M (that is, m ∈ N ⇒ m ∈ M ), then N is called a subset of M (we write N ⊂ M or N ⊆ M ). The symbol ⇒ is used in place of from which it follows that. Two sets M 1 and M 2 are equal (we write M 1 = M 2 ), when M 1 ⊂ M 2 and M 2 ⊂ M 1. That means m ∈ M 1 ⇔ m ∈ M 2. Here the symbol ⇔ is written instead of if and only if, that is both ⇒ as well as ⇐ holds.

Example 1.1.5 { 0 , ..., 9 } ⊂ N 0.

Definition 1.1.6 Let M, N be sets. Then the complement of N in M is, M ƒN = {m ∈ M S m ∉ N }.

The complement M ƒN can be seen in terms of a so-called Venn- Diagram in Figure 1.1. Further we call

M ∪ N = {m S m ∈ M or m ∈ N }

the union of M and N , see Figure 1.2, and

M ∩ N = {m S m ∈ M und m ∈ N }

intersection of M and N , see Figure 1.3.

1. FUNDAMENTAL CONSTRUCTIONS 6

Figure 1.1: Complement

Figure 1.2: Union

Example 1.1.7 N 0 = N ∪ { 0 }.

Notation 1.1.8 For an index set I ≠ ∅ and sets Mi, i ∈ I, we write

 i∈I

Mi = {m S m ∈ Mi for all i ∈ I}

for the intersection of the Mi, i ∈ I, and

 i∈I

Mi = {m S there exists i ∈ I with m ∈ Mi}

Figure 1.3: Intersection

1. FUNDAMENTAL CONSTRUCTIONS 8

Definition 1.1.14 Let M be a set. The power set of M is

2 M^ = P(M ) = {A S A ⊂ M }.

Theorem 1.1.15 Let M be a finite set. Then

T 2 M^ T = 2 SM^ S.

Example 1.1.16 Power sets:

2 ∅^ = {∅} 2 {^1 }^ = {∅, { 1 }} 2 {^1 ,^2 }^ = {∅, { 1 }, { 2 }, { 1 , 2 }}.

We use the following general principle for proofs to prove, for example, Theorem 1.1.15.

1.2 Mathematical induction

Suppose we have for every n ∈ N 0 a given claim A(n), and fur- thermore it is given that:

1. Base case: A( 0 ) is true.

2. Induction step: it follows for every n > 0 that

A(n − 1 ) is true ⇒ A(n) is true.

Then A(n) is true for all n ∈ N 0. In fact we have the following chain of conclusions:

A( 0 ) true ⇒ A( 1 ) true ⇒ A( 2 ) true ⇒ ...

Remark 1.2.1 Analogously, one can of course proceed to prove statements A(n) for n ≥ n 0 with n 0 ∈ Z. One only has to make sure that the initial step A(n 0 ) and all subsequent arrows used in the chain of conclusions

A(n 0 ) true ⇒ A(n 0 + 1 ) true ⇒ A(n 0 + 2 ) true ⇒ ...

are proven.

1. FUNDAMENTAL CONSTRUCTIONS 9

Using induction, we now prove Theorem 1.1.15: Proof. By numbering the elements of M we can assume without loss of generality (written in short WLOG) that M = { 1 , ... , n}, using the convention that { 1 , ... , 0 } = ∅. So we have to show that the statement T 2 {^1 ,...,n}T = 2 n

hold true for all n ∈ N 0. Initial step n = 0 : It is 2 ∅^ = {∅}, and hence S 2 ∅S = 1 = 20. Inductive step n − 1 to n: The union

2 {^1 ,...,n}^ = {A ⊂ { 1 , ..., n} S n ∉ A}

⋅ ∪ {A ⊂ { 1 , ..., n} S n ∈ A} = {A S A ⊂ { 1 , ..., n − 1 }}

⋅ ∪ {A′^ ∪ {n} S A′^ ⊂ { 1 , ..., n − 1 }}

is disjoint, and therefore it follows from the induction hypoth- esis T 2 {^1 ,...,n−^1 }T = 2 n−^1 ,

that T 2 {^1 ,...,n}T (^) = 2 n−^1 + 2 n−^1 = 2 n.

Next we discuss another typical example of a proof using induction.

Notation 1.2.2 We write

n Q k= 1

ak = a 1 + ... + an

for the sum of the numbers a 1 , ..., an. Similar we use (^) n M k= 1

ak = a 1 ⋅ ... ⋅ an

for their product. When the numbers ak are represented by the elements k of the set I, we write Q k∈I

ak

for their sum and, analoguesly ∏k∈I ak for their product.

1. FUNDAMENTAL CONSTRUCTIONS 11

Remark 1.2.5 The analogue to a proof by induction is in com- puter science the concept of a recursive algorithm. For example, the following recursive function calculates the sum (^) ∑nk= 0 k:

sumints:=proc(n) if n=0 then return(0);fi; return(sumints(n-1)+n); end proc; We can also write a recursive function that determines all subsets of { 1 , ..., n} from the proof of Theorem 1.1.15. For the implementation thereof, see Exercise 1.8. Another proof by in- duction, which provides a recursive algorithm, is discussed in Exercises 1.10 and 1.11.

For further examples of induction, see the Exercises 1.5, 1.6, 1.7 and 1.12.

1.3 Relations

In the following way we can describe relations between two sets:

Definition 1.3.1 A relation relation between sets M and N is given by the subset R ⊂ M × N.

Example 1.3.2 For M = { 2 , 3 , 7 }, N = { 4 , 5 , 6 } and

R = {(m, n) ∈ M × N S m divides n}

we have R = {( 2 , 4 ), ( 2 , 6 ), ( 3 , 6 )}.

The most important role is played by relations in which each element of M gets assigned exactly one element of N :

1.4 Maps

Definition 1.4.1 A map f ∶ M → N is a relation R ⊂ M × N , such that for every m ∈ M there is a unique element f (m) ∈ N with (m, f (m)) ∈ R. We write

f ∶ M → N m ↦ f (m).

1. FUNDAMENTAL CONSTRUCTIONS 12

We call M the source and N the target of f. For a subset A ⊂ M

f (A) = {f (m) S m ∈ A} ⊂ N

is called the image of A under f , and

Image(f ) ∶= f (M )

is called the image of f. For B ⊂ N

f −^1 (B) = {m ∈ M S f (m) ∈ B} ⊂ M

is called the preimage of B under f.

Remark 1.4.2 If a map is given by a mapping rule f ∶ M → N , m ↦ f (m), the representation of f as a relation is nothing else than the graph

R = Graph(f ) = {(m, f (m)) S m ∈ M } ⊂ M × N

of f.

Example 1.4.3 For

f ∶ R → R x ↦ f (x) = x^2

we have R = Graph(f ) = ™(x, x^2 ) S x ∈ Rž ,

see Figure 1.4. The image of f is

f (R) = R≥ 0

and we have

f −^1 ({ 1 , 2 }) = {− 1 , 1 , −

Definition 1.4.4 A map f ∶ M → N is surjective, if for the image of f we have f (M ) = N. If for all m 1 , m 2 ∈ M we have, that

f (m 1 ) = f (m 2 ) Ô⇒ m 1 = m 2 ,

then f is injective. A map that is both injective and surjective is bijective.