Algebra

Question 1

The Division Algorithm

Answer 1

The Division Algorithm states that for any integers a and b (with b>0), there exist unique integers
q (quotient) and r (remainder) such that:

a = bq +r
and
0<=r<b

This pair(q,r) is unique for given a and b

Question 2

Definition of Divisibility

Answer 2

An integer a divides another integer b if there exists an integer
c such that b=ac. This relationship is often denoted as a∣b.

Question 3

What integer can 0 divide?

Answer 3

The only integer that 0 divides is 0 itself.

Question 4

Definition of Greatest Common Divisor (GCD)

Answer 4

The GCD of two integers
a and b is a non-negative integer
r that divides both a and b and is greater than any other common divisor s of a and b.

Question 5

Divisibility of the GCD

Answer 5

If r is the greatest common divisor (gcd) of integers a and
b, then s divides r

Question 6

GCD with Zero

Answer 6

For any non-negative integer
a, the greatest common divisor (gcd) of a and 0 is a

Question 7

GCD Properties with Sign Changes

Answer 7

The greatest common divisor (gcd) is unaffected by the signs of its arguments
i.e. gcd(a,b) = gcd(-a,-b)

Question 8

Bézout’s Identity

Answer 8

For any integers a and b, there exist integers r and s such that
ar+bs=gcd(a,b)

Question 9

Euclid’s Algorithm Foundation

Answer 9

For any integers a and b (where b>0), there exist unique integers
q (quotient) and r (remainder,
0≤r<b) such that a=bq+r.

Then gcd(a, b) = gcd(b,r)

Question 10

Definition of a Prime Number

Answer 10

A prime number is a positive integer n whose positive integer divisor is 1 or itself.

By Bezout, If n divides the product ab of two integers a and b, then n must divide at least one of a or b.

This is used to prove the FTA

Question 11

Fundamental Theorem of Arithmetic

Answer 11

Every integer can be expressed uniquely as a product of primes raised to their respective powers, often written as:
(−1) ^r∞ ∏ p^rp

1. p represents a prime number,
2. rp is the non-negative integer exponent of the prime p.
3. r∞ is 0 or 1, indicating the sign (1 for negative numbers, 0 for non-negative numbers).
   ​

Question 12

Equivalence Relations

Answer 12

Let R be a relation on S. For an element a in S, the equivalence class [a] includes all elements b in
S that are related to a under
R, denoted as aRb.

If R is an equivalence relation, then

aRb if and only if [a] = [b]

And Equivalence relation is said to be reflexive, symmetric and transitive.

Question 13

Bijective Correspondence between Equivalence Relations and Partitions

Answer 13

Given a set S, there exists a bijective correspondence between
* the equivalence relations R on S,
* the partitions P (a set of non-overlapping subsets of S) on S.

Question 14

Equivalence Relation for Congruence Modulo n

Answer 14

For a positive integer n, the relation R on Z (set of all integers) defined by a≡b(modn) forms an equivalence relation if and only if n divides b − a

Question 15

Definition of Zn

Answer 15

Let Zn denote the set of equivalence classes [a] with respect to (≡, Z).

Since a ≡ b mod n if and only if
[a] = [b], a lot of equivalence classes may be identified

Question 16

Operations in Zn and Size of Zn
​

Answer 16

Size of Zn:|Zn| = n
Addition: [a]+[b]=[a+b]
Subtraction: [a]−[b]=[a−b]
Multiplication: [a][b]=[ab]
Division: Not defined

Equivalence Under Operations: If a≡a′ modn and b≡b′ modn, then
[a]+[b]=[a′]+[b′], [a]−[b]=[a′]−[b′] and [a][b]=[a′][b′]

Question 17

Multiplicative Inverse in Zn
​

Answer 17

An element [a] in Zn has a multiplicative inverse if there exists integer b such that [a][b]=[1] (equivalent to ab≡1modn).

Multiplicative inverse of [a] is denoted as [a]^−1

Question 18

Uniqueness of Multiplicative Inverse in Zn

Answer 18

If [a] in Zn has a multiplicative inverse, it is unique.

Given two elements [b] and [c] such that [a][b]=[1] and [a][c]=[1],
By associativity, c=[c][1] simplifies to ([c][a])[b]=[1][b], showing [c]=[b]

Question 19

Condition for Multiplicative Inverse in Zn

Answer 19

An element [a] of Zn has a multiplicative inverse if and only if
gcd(a,n)=1.

This is verified using Euclid’s algorithm, which finds integers
b and c such that ab+nc=gcd(a,n)=1.

Thus, ab≡1modn, confirming
[a][b]=[1]

Question 20

Absence of Multiplicative Inverse Zn
​

Answer 20

An element [a] of Zn lacks a multiplicative inverse if and only if there is a non-zero b such that ab≡0 modn

Question 21

Group Definition

Answer 21

A group is a set G equipped with an operation ∗ that satisfies the following axioms:

(G0) Closure: For all a,b in G, the result of a∗b is also in G

(G1) Associativity: For all a,b,c in G,
(a∗b)∗c=a∗(b∗c)

(G2) Identity: There exists an element e in G such that
a∗e=e∗a=a for every a in G

(G3) Inverses: For every element
a in G, there exists an element
b in G such that a∗b=b∗a=e, where
e is the identity element

(G4) Commutativity (For abelian groups): For all a,b in G, a∗b=b∗a

When these five conditions hold, we say (G, ∗) (or simply G if the operation ∗ is clear from the
context) is a commutative/abelian group

Question 22

Basic Properties of a Group (G, * )

Answer 22

Uniqueness of Identity: The identity element of G is unique

Uniqueness of Inverses: Each element a in G has a unique inverse, denoted as
a^−1

Cancellation Law: If a∗b=a∗c, then b=c. Similarly, b∗a=c∗a, then b=c

Inverse of a Product: For any a,b in G, the inverse of their product is (a∗b)^−1=b^−1∗a^−1

Question 23

Ring Axioms

Answer 23

A ring R is a set equipped with two operations: addition (+) and multiplication (×), which satisfy the following axioms:

Addition Axioms:
R+0: Closure under addition.
R+1: Associativity of addition.
R+2: Existence of additive identity (0).
R+3: Existence of additive inverses.
R+4: Commutativity of addition.

Multiplication Axioms:
Rx0: Closure under multiplication.
Rx1: Associativity of multiplication

Distributive Axioms:
Rx+: Distributivity of multiplication over addition from the left.
R+x: Distributivity of multiplication over addition from the right.

Question 24

Abelian Group from Ring Axioms

Answer 24

The Addition axioms say that
(G,*) = (R,+) is an additive (abelian group.

Question 25

Operations in Groups and Rings

Answer 25

In groups and rings, the operations + and × are conventional symbols for the operations that meet specific conditions.

Question 26

Notation for Multiplication in Algebra

Answer 26

We often write ab instead of a × b

Question 27

Definition of a Commutative Ring

Answer 27

A ring R is said to be a commutative ring if a × b = b × a holds for all a, b in R

Question 28

Additive Properties of a Ring

Answer 28

Unique Zero Element: Every ring
R is equipped with a unique zero element that acts as an additive identity; that is, a+0=0+a=a for any a in R.

Unique Additive Inverse: Each element a in R has a unique additive inverse −a such that
a+(−a)=0

Cancellation Law: If a+b=a+c within R, then b=c.

Question 29

Zero Product Property in a Ring

Answer 29

Let R be a ring. For every element a of R, we have 0a = a0 = 0

Question 30

When does a Ring have an Identity and what are they

Answer 30

Ring with Identity: A ring R is said to have an identity if there exists an element 1 in R such that for every a in R, a×1=1×a=a

The additive identity 0 and the multiplicative identity
(if exists) do not have to be distinct

Question 31

Is Zn as a Commutative Ring?

Answer 31

The set Zn, with addition and multiplication modulo n as defined before, is a
commutative ring with identity [1].

Question 32

Definition of a Unit in a Ring

Answer 32

Let R be a ring with identity element 1. An element a in R is called a unit if there
is an element b in R such that ab = ba = 1. The element b is called the inverse of a.

Question 33

What are the Units in a Ring with Identity

Answer 33

{units in R} = {elements in R that have multiplicative inverses}

Question 34

Notation for Units in a Ring

Answer 34

The notation R^× is used to denote the units of a ring R

Question 35

Units of Zn

Answer 35

The units of Zn are the equivalence classes [a] where
gcd(a,n)=1

Furthermore, |Zn| = φ(n)= Product (p-1)p^rp-1

Question 36

Rings with Multiplicative Identities of 1 (units)

Answer 36

The identity element 1 is unique.

If 1 is distinct from the additive identity 0, then 0 is NOT a unit.

1 is a unit and its inverse is 1 itself

Question 37

Properties of Units in Rings

Answer 37

If a is a unit, the inverse of a is unique.

If a is a unit, then so is a^−1 the inverse of a

If a and b are units, then so is ab; and its inverse is b^−1a^−1

Question 38

Set of Units R^x

Answer 38

For a ring R with identity, the set
R^× of units forms a group under multiplication

If the ring R is commutative, the group R^× is abelian,

Question 39

Definition of a Field

Answer 39

A field is a commutative ring
(F, +, ×) satisfying the axioms:

(F, +) is an (abelian) additive group (with identity element 0)

(F −{0}, ×)is a multiplicative group (with identity element 1)

The additive identity ‘0’ (the identity element in the group
(F, +)) is distinct from the
multiplicative identity ‘1’ (the identity element in the group
(F − {0}, ×))

Question 40

When do we deny a set from becoming a field and why?

Answer 40

If in a ring the multiplicative identity (1) is equal to the additive identity (0), then every element
a in the ring would satisfy 1×a=0×a=0

So the condition 1 /= 0 denies any set with one element {1 = 0} any chance of being a field

Question 41

Hierarchical Structure of Algebraic Systems

Answer 41

Field ⇒ Ring ⇒ Group

Question 42

Field Structure of Fp

Answer 42

If p is a prime number, then Fp = Zp is a field

Question 43

The set of Complex numbers

Answer 43

Complex numbers are defined as elements of the form a+bsqrt(−1)
​, where a and b are real numbers. The set of complex numbers is denoted by C

Question 44

Field Structure of Complex Numbers

Answer 44

The set C is a field

Question 45

Division Ring

Answer 45

We say that a ring R with identity is called a division ring if it satisfies all the axioms except the commutativity of multiplication (a × b = b × a for all a, b in R)

a field assumes the set of non-zero elements is an abelian group with respect to ×

Question 46

Cancellation Law in Division Rings

Answer 46

Let R be a division ring and a is non-zero element of R. If ab = ac, then b = c

Question 47

Definition of Polynomials

Answer 47

A polynomial f in one variable X with coefficients in a ring R is expressed as
f=cnX^n+cn−1X^n−1+⋯+c1X+c, where c n ,cn−1,…,c1 ,c are elements of R, known as the coefficients of f.

The set of all such polynomials is denoted by R[X]

Question 48

Degree of a Polynomial

Answer 48

For a non-zero polynomial f in one variable X, the degree of f, denoted as deg(f), is the highest power n for which the coefficient
cn of Xn is non-zero

Question 49

Monic Polynomials

Answer 49

A non-zero polynomial f = cnX^n + cn−1X^n−1 + · · · + c1X + c of degree n is called monic if the leading coefficient cn = 1.

The zero polynomial is defined to be monic

Question 50

Operations and Properties of Polynomial Rings

Answer 50

Addition: (f + g)(X) = f(X) + g(X) =
SUMn (cn(f) + cn(g))X^n

Multiplication: (fg)(X) = f(X)g(X) = SUMn ( SUMr cr(f)c(n−r)(g)

Properties:

If R is a ring, then so is R[X] in terms of addition

If R is a ring with identity, then so is R[X]

If R is commutative, then so is R[X]

Question 51

Polynomial Rings and Division

Answer 51

If (R, +, ×) is a ring with identity 1, then R[X] is not a division ring

Question 52

Units in Polynomial Rings

Answer 52

Let (F, +, ×) be a field. The units F[X]× of F[X] are F× = F − {0}

Question 53

Division Algorithm for Polynomials

Answer 53

For any polynomials f and g in
F[X] where g is non-zero, there exist polynomials q (quotient) and
r (remainder) in F[X] such that:

f=gq+r

where r=0 or deg(r)<deg(g)

Question 54

Divisibility Among Polynomials

Answer 54

Let f and g be polynomials in F[X]. We say that g divides f , or g is a factor of f ,if there exists a polynomial q in F[X] such that f = gq

Question 55

Polynomial Factorization in Fields

Answer 55

Let F be a field. Let f in F[X] and α be an element of F. Then there exists q in F[X] and r in F such that
f = (X − α)q + r

Question 56

Evaluating Roots and Factors of Polynomials

Answer 56

For any polynomial f in F[X] and any a in the field F, the value
f(a) represents the remainder when f is divided by (X−a)

Question 57

Fundamental Theorem of Algebra in Complex Polynomials

Answer 57

For any non-zero polynomial of degree n≥1 with coefficients in the complex numbers C, 𝑐𝑛𝑋^𝑛+…+𝑐0
​must have at least one root within
C.

Question 58

The Complex Fundamental Theorem of Algebra in Polynomials with multiplicities

Answer 58

Every non-zero polynomial f(X)=cn
​X^n+…+c0 of degree n≥1 with coefficients in the complex numbers C can be completely factored into n roots in C, including their multiplicities.

f(X) = cn(X − αn)(X − αn−1)…(X − α1)

where α1,…,αn are the roots of f(X)

Question 59

Generalizing GCD for Polynomials

Answer 59

Any two polynomials f and g have a greatest common divisor in F[X].

The gcd of two polynomials in F[X] can be found by Euclid’s algorithm

If gcd(f , g) = γ (a polynomial in F[X]), then there exist p, q in F[X] such that
fp + gq = γ

Question 60

Ring Structure of 2x2 Matrices

Answer 60

M2(R) is a ring. If R is a ring with identity, then so is M2(R)

Question 61

Additive and Multiplicative Identities in Matrix Rings

Answer 61

The additive identity in a matrix ring M2(R) is represented by a 2x2 zero matrix

The multiplicative identity, assuming R has an identity element 1, is the 2x2 identity matrix

Question 62

Non-Commutativity of 2x2 Matrix Rings

Answer 62

M2(R) is never commutative, even if R is commutative

Question 63

Properties of Non-Trivial Matrix Rings

Answer 63

For a ring R with identity, if for
every elements a, b in R, the product is always ab = 0, then M2(R) is neither commutative nor a division ring.

Question 64

Permutation of a Set

Answer 64

A permutation of a set S is a function f : S –> S which is bijective (one-to-one and
onto)

Question 65

Notation for Permutations

Answer 65

The set of permutations on the set {1, . . . , n} is denoted Sn

every element σ in Sn is written as:
f = (1 … n)
(f(1) … f(n))

Question 66

Size of the set of permutations

Answer 66

|Sn| = n!.

Question 67

Composition of permutations

Answer 67

If f and g are permutations, we define the composition, denoted
f ◦ g to be that which sends s in S to f(g(s))

Question 68

Set of the composition of permutations

Answer 68

If f and g are elements of Sn, then so is the composite f ◦ g is in Sn

Question 69

Inverse of permutations

Answer 69

If f is in Sn, then the inverse function f^−1 exists and is an element of Sn

Question 70

Order of permutations

Answer 70

Let f be an element of Sn. The order of f is the smallest number of times we compose f with f itself, f ◦ f ◦ f … , to get the identity

Question 71

Cycles in Permutation Groups

Answer 71

A cycle (y1 ,y2 ,…,yr) in the symmetric group Sn represents a permutation that sends each element yi to yi+1 (with yr mapping back to y1), while all other elements remain unchanged. This transformation maps the sequence as y1→y2
→…→yr→y1, leaving all elements not in {y1 ,…,yr} unchanged.

Question 72

Disjoint notation of Cycles

Answer 72

Any permutation can be written as a composition of disjoint cycles. The representation is unique, up to the facts that:

The cycles can be written in any order,

Each cycle can be started at any point,

Cycles of length 1 can be left out.

Question 73

Calculating the order of a permutation

Answer 73

The order of a permutation is the least common multiple of the lengths of the
cycles in the disjoint cycle representation.

Question 74

Permutations as Groups

Answer 74

(Sn, ◦(composition)) is a group

Question 75

When is the Group of Permutations abelian

Answer 75

Sn is an abelian group if n 6 2 and is non-abelian if n > 2

Question 76

Definition of a Subgroup

Answer 76

Let (G, ∗) be a group and Γ be a subset. We say that Γ is a subgroup of G if (Γ, ∗) is a group.

(G0) If a and b are elements of Γ, then a ∗ b is an element of Γ

(G1) If a, b,c are elements of Γ, then (a ∗ b) ∗ c = a ∗ (b ∗ c) holds.

(G2) Γ contains the identity element eΓ. In fact, eΓ = e (the identity element of G)

(G3) Every element of Γ has an inverse. By the uniqueness, this inverse is the inverse we get when
we think of it as an element of G

Question 77

Lagrange’s Theorem

Answer 77

Let G be a finite group and H be a subgroup. Then |H|
divides |G|

Question 78

When is Γ a subgroup of (G, *)

Answer 78

A non-empty subset Γ of a group (G, ∗) is a subgroup if and only if, for every g, γ in Γ, g ∗ γ^−1 is in Γ.

Question 79

Properties of a Partition

Answer 79

∅ is not a part of P

If A and B are distinct parts of P, then A ∩ B = ∅,

The union of all parts of P is S