NOUN e-Exams Past Questions & Answers: MTH211 - Abstract Algebra

Email: Solutions2tma@gmail.com

Whatsapp: 08155572788

Multiple Choice Questions (MCQs)

MCQ1

In a principle ideal Domain an element is prime if and only if it is

Irreducible

0.0000000

Reducible

1.0000000

Even

0.0000000

odd

0.0000000

MCQ2

Let R be an integral domain. We say that an element x ∈ R is irreducible if

(I) x is not a unit

(II) If x = ab with a,b ∈ R then a is a unit or b is a unit.

Which of the following is the definition of irreducible element

I only

1.0000000

II only

0.0000000

I and II

0.0000000

None of the option

0.0000000

MCQ3

In Qx find the g.c.d of p(x) = x2+3x-10 and q(x) = 6x2-10x-4

x-2

0.0000000

X+5

0.0000000

3x+1

1.0000000

None of the option

0.0000000

MCQ4

An element d ∈ R is a greatest common divisor of a,b ∈ R if

I d/a and d/b

II For any common divisor c of a and b, c/d which of the following is a properties of greatest common divisor

I only

0.0000000

II only

1.0000000

I and II

0.0000000

None of the option

0.0000000

MCQ5

Let R be an integral domain. We say that a function d: R\{0} N ∪ {0} is a Euclidean valuation on R if which of the following conditions are satisfied:

I d(a) ≤ d(ab) ∀ a,b ∈ R\{0}

II for any a,b ∈ R, b ≠ 0 ∃ q, r ∈ R such that a = bq + r where r = 0 or d(r) < d(b)

I only

0.0000000

I and II

0.0000000

II only

1.0000000

None of the option

0.0000000

MCQ6

Let p be a prime number consider xp-1-T∈ ZP [x]. Use the fact that ZP is a group of order p. show that every non – zero element of ZP is a root of xp-1-T. In particular if p = 3

x3-1-T = (x – T)(x – )

1.0000000

x3-1-T = (x + )(x + )

0.0000000

x3-1-T = (x + )(x + )

0.0000000

None of the option

0.0000000

MCQ7

In the given polynomial f(x) = x-32(x+2), 3 is a root of multiplicity

1

1.0000000

2

0.0000000

0

0.0000000

None of the option

0.0000000

MCQ8

Let F be a field and f(x) ∈ F[x]. We say that an element a∈F is a root of f(x) if

f(a) ≠ 0

0.0000000

f(a) = 1

1.0000000

f(a) = 0

0.0000000

None of the option

0.0000000

MCQ9

Express x4+ x3+5x2-x as (x2 +x+1)+rx in Q[x]

x4+ x3+5x2-x = x2+ x+1x2+ 4-(5x+4)

0.0000000

x4+ x3+5x2-x = x2+ x+1x+ 4-(5x+4)

0.0000000

x4+ x3+5x2-x = x2+ x+1x2- 4-(5x+4)

0.0000000

None of the option

1.0000000

MCQ10

Let F be a field. Let f(x) and g(x) be two polynomials in F[x] with g(x) ≠0. Then

I There exist two polynomial q(x) and r(x) in F[x] such that f(x) = q(x)g(x) + r(x), where degr(x) < degg(x).

IIThe polynomial q(x) and r(x) are unique, which of the following is a properties of Division Algorithm

I only

1.0000000

II only

0.0000000

I and II

0.0000000

None of the option

0.0000000

MCQ11

Which of the following polynomial ring is free from zero divisor

Z6

1.0000000

Z7

0.0000000

Z4

0.0000000

Z8

0.0000000

MCQ12

Let R be a ring and f(x) and g(x) be two non – zero element of R[x]. Then deg(f(x)g(x)) ≤ degf(x) + degg(x) with equality if

R has a zero divisor

0.0000000

R is an integral domain

0.0000000

R does not have a zero divisor

1.0000000

None of the option

0.0000000

MCQ13

If p(x), q(x) ∈ Z[x] then the deg(p(x).q(x)) is

Deg p(x) + deg q(x)

0.0000000

Max (deg p(x), deg q(x))

1.0000000

Min (deg p(x), deg q(x))

0.0000000

None of the option

0.0000000

MCQ14

If f(x) = a0+a1x+…+anxn and g(x) = b0+b1x+…+bmxm are two polynomial in R[x], we define their product f(x).g(x) = c0+c1x+…+cm+nxm+1 where ci is

ai bi ∀ i = 0,1, …, m+n

1.0000000

ai b0 ∀ i = 0,1, …, m+n

0.0000000

ai b0+ ai+1 b1+…+a0 bi ∀ i = 0,1, …, m+n

0.0000000

None of the option

0.0000000

MCQ15

Consider the two polynomials p(x), q(x) in Z[x] by p(x) = 1+2x+3x2, q(x) = 4+5x+7x3. Then p(x) + q(x) is

4+7x+3x2+7x3

0.0000000

5+7x+3x2+7x3

1.0000000

1+7x+3x2+7x3

0.0000000

None of the option

0.0000000

MCQ16

Determine the degree and the leading coefficient of the polynomial 1+x3+x4+0.x5 is

(4,1)

0.0000000

(3,1)

1.0000000

(5,1)

0.0000000

(5,0)

0.0000000

MCQ17

The Degree of a polynomial written in this form deg(∑i=0naixi) if an ≠0 is

0

1.0000000

n

0.0000000

i

0.0000000

None of the option

0.0000000

MCQ18

Let R be a domain and x ∈ R be nilpotent then xn = 0 for some n ∈ N. Since R has no zero divisors this implies that

x = 0

0.0000000

x = 1

1.0000000

x = 2

0.0000000

None of the option

0.0000000

MCQ19

An ideal m Z of Z is maximal if and only if m is

An even number

1.0000000

An odd number

0.0000000

A prime number

0.0000000

None of the option

0.0000000

MCQ20

Every maximal ideal of a ring with identity is

A prime ideal

0.0000000

A field

1.0000000

An integral domain

0.0000000

None of the option

0.0000000

MCQ21

Let R be a ring with identity. An ideal M in R is Maximal if and only if R/M is

An ideal

0.0000000

A field

1.0000000

An integral domain

0.0000000

None of the option

0.0000000

MCQ22

An ideal p of a ring R with identity is a prime ideal of R if and only if the quotient ring

An integral domain

1.0000000

An ideal

0.0000000

Zero ideal

0.0000000

None of the option

0.0000000

MCQ23

The characteristics of a field is either

Zero or even number

0.0000000

Zero or prime number

0.0000000

Zero or odd number

0.0000000

None of the option

1.0000000

MCQ24

Zn is a field if and only if

n is an even number

1.0000000

n is an old number

0.0000000

n is a prime number

0.0000000

None of the option

0.0000000

MCQ25

Which of the following is an axioms of a field

Is commutative

1.0000000

R has identity (which is denoted by I) and I ≠ 0

0.0000000

Every non – zero element x ∈ R has a multiplicative inverse which we denote by x-1

0.0000000

All of the option

0.0000000

MCQ26

Let R be a ring, the least positive integer n such that nx = 0 ∀ x ∈ R is called

Characteristics of R

0.0000000

The order of R

1.0000000

The value of R

0.0000000

None of the option

0.0000000

MCQ27

Which of the following is not a property of an integral domain

Is a commutative ring

1.0000000

Is with unity element

0.0000000

Does not contain a zero divisor

0.0000000

None of the option

0.0000000

MCQ28

A non – zero element in a ring R is called zero divisor in R if there exist a non – zero element b in R such that

ab ≠ 0

0.0000000

ab = 0

1.0000000

ab-1 = 0

0.0000000

None of the option

0.0000000

MCQ29

If H is a subgroup of a group G and a, b ∈ G then which of the following statement is true?

aH = H Iff a∈ H

0.0000000

Ha = H Iff a ∈ H

1.0000000

Ha = Hb Iff a-1a ∈ H

0.0000000

All the option

0.0000000

MCQ30

Let G be a group and a∈G such that O(G) = t, then an= am, if and only if

n ≡ m (mod t)

0.0000000

n ≡ t (mod n)

0.0000000

m ≡ t (mod n)

0.0000000

None of the option

1.0000000

MCQ31

Which of these does not hold for ‘×’ distributive over∪, ∩ and ‘ –

A× (B∪C) = A×B ∪ A×C

1.0000000

A× (B∩C) = A×B ∩ A×C

0.0000000

A× (B – C) = A×B – A×C

0.0000000

None of the above

0.0000000

MCQ32

The symmetric difference of two given sets A and B, denoted by A ∆ B is defined by

A ∆ B = ( A – B) ∩ (B – A)

0.0000000

A ∆ B = ( A – B) ∪ (B – A)

0.0000000

A ∆ B = ( A – B) or (B – A)

1.0000000

None of the above

0.0000000

MCQ33

The (relative) complement (or difference) of a set A with respect to a set B denoted by B – A (or B\A) is the set

B – A = {x∈ B A}

0.0000000

B – A = {x B A}

0.0000000

B – A = {x B ∈A}

1.0000000

None of the option

0.0000000

MCQ34

Which of the following is of the operations ∪and ∩

Idempotent : A ∪A = A = A∩A for every set A

0.0000000

Associative A ∪ (B ∪C) = (A ∪B) ∪C and A∩ (B∩C) = (A∩B) ∩ C for three sets A,B,C

1.0000000

Commutative: AB = B ∪A and A∩B = B∩A for any two sets A, B

0.0000000

All the option

0.0000000

MCQ35

The intersection of two sets A and B written as A∩B is

The set A∩B = {x:x∈A and x∈B}

1.0000000

The set A∩B = {x:x∈A or x∈B}

0.0000000

The set A∩B = {x:x∈A and x ∉B}

0.0000000

The set A∩B = {x:x∈A or x ∉B

0.0000000

MCQ36

A set X of n elements has

n subsets

0.0000000

2n subsets

1.0000000

2 subsets

0.0000000

All the option

0.0000000

MCQ37

If G is a finite group such that O(G) is neither I nor a prime, then G has

Non – trivial proper subgroup

1.0000000

Trivial proper subgroup

0.0000000

Subgroup of order prime

0.0000000

Non – trivial subgroup of order prime

0.0000000

MCQ38

Which of the following is not the definition of Euler Phi – function ϕ :Ν ⟶ Ν

ϕ (i=1(

1.0000000

ϕ x= number of natural numbers less than n and relatively prime to n

0.0000000

ϕ x= number of natural numbers greater than n and relatively prime to n

0.0000000

None of the option

0.0000000

MCQ39

Every group of prime order is

Non – abelian

1.0000000

Cyclic

0.0000000

Distinct

0.0000000

All the option

0.0000000

MCQ40

An element is of infinite order if and only if all its power are

Real

1.0000000

Imaginary

0.0000000

Distinct

0.0000000

None of the above

0.0000000

MCQ41

Consider the following set of 8 2 ´ 2 matrices over ¢. Q8 = {±I, ±A, ±B, ±C}

where I = 1001, A = 01-10, B =0i0-i, C = i00-i and i = -1. If H = <A> is a subgroup, how many distinct right cosets does it have in Q8

2

0.0000000

4

0.0000000

8

1.0000000

6

0.0000000

MCQ42

Let H = 4Z. How many distinct right coset of H in Z do we have?

2

1.0000000

4

0.0000000

6

0.0000000

8

0.0000000

MCQ43

A function f : A ⟶B is called one – one if and only if different element of B. some time is called

Surjective

0.0000000

Injective

0.0000000

Bijective

1.0000000

None of the above

0.0000000

MCQ44

Let G be a group, g ∈ G and m, n ∈ Z. which of the following does not hold

gmg-m = e that is g-m = (gm)-1

0.0000000

(gm)n = gmn

1.0000000

gmgn=gm+n

0.0000000

None of the above

0.0000000

MCQ45

Let G be a group. If there exist g ∈ G has the form x = gn for some n ∈ Z then G is

A cyclic group

1.0000000

A noncyclic group

0.0000000

An infinite group

0.0000000

All the option

0.0000000

MCQ46

Let H = {I, (1, 2)} be a subgroup of S3. The distinct left cosets of H in S3are

H, (13)H, (23)H

0.0000000

H, (123)H, (12)H

1.0000000

H, (132)H

0.0000000

None of the option

0.0000000

MCQ47

The order of 01-10 in Q8 is

0

0.0000000

2

0.0000000

4

1.0000000

6

0.0000000

MCQ48

The order of (12) in S3 is

1

1.0000000

2

0.0000000

3

0.0000000

4

0.0000000

MCQ49

A group generated by g is given by <g> = {e, g, g2, …,gm-1} the order of g is

M

0.0000000

M-1

0.0000000

0

1.0000000

2

0.0000000

MCQ50

Let H be a subgroup of a finite group G. We call the number of distinct of H in G is

Order of subgroup

0.0000000

index

1.0000000

Order of the group

0.0000000

Order of an element

0.0000000

Fill in the Blank (FBQs)

FBQ1

Let G = {1, -1, i, -i}. Then G is a group under usual multiplication of complex numbers, in this group, the order of i is ______________.

*4*

1.0000000

0.0000000

FBQ2

The degree and the leading coefficient of the polynomial 1 + x3+x4+0.x5is _____________________.

*(4,1)*

1.0000000

0.0000000

0.0000000

FBQ3

The degree of a polynomial written in this form (∑i=0naixi) if an≠0 is_____________________.

*n*

1.0000000

0.0000000

FBQ4

The order of (12) in S3is ___________________.

*2*

1.0000000

0.0000000

FBQ5

In a permutation, any cycle of length two is called __________________.

*Transposition*

1.0000000

0.0000000

FBQ6

A field K is called _____________ of F if F is a subfield of K, thus Q is a subfield of R and R is a field extension of Q

*Field extention*

1.0000000

0.0000000

FBQ7

A non – empty subset S of a field F is called a subfield of F if it is a field with respect to the operations on F. if S ≠F, then S is called ____________ of F

*Proper subfield*

1.0000000

0.0000000

FBQ8

Let f(x) = a0+a1x+…anxn∈Zx. We define the content of fx to be the g.c.d of the integers a0,a1,…,an, we say f(x) is _______________ if the content of f(x) = 1

primitive

1.0000000

0.0000000

FBQ9

We call an integral domain R a _______________ if every non – zero element of R which is not a unit in R can be uniquely expressed as a product of a finite number of irreducible elements of R

*Unique factorization domain*

1.0000000

0.0000000

FBQ10

An element d ∈ R is a ________________ of a, b ∈ R if

d|a and d|b and (i)i for any common divisor c of a and b, c|d

*Greatest Common divisor*

1.0000000

0.0000000

FBQ11

Given two elements a and b in a ring R, we say that c ∈ R is a ______________ of a and b if c|a and c|b.

*Common divisor*

1.0000000

0.0000000

FBQ12

We call an integral domain R a ________________ if every ideal in R is a principal ideal.

*Principal ideal*

1.0000000

0.0000000

FBQ13

The number of unit that can be obtained in R = a+b-5 |a,b ∈Z is ______________

*2*

1.0000000

0.0000000

FBQ14

Let R be an integral domain, an element a ∈ R is called a unit or an ____________ in R if we can find b∈R such that ab = 1 i.e if a has a multiplicative inverse

*Invertible element*

1.0000000

0.0000000

FBQ15

A domain on which we can define a Euclidean valuation is called ________.

*Euclidean domain*

1.0000000

*Euclidean*

1.0000000

FBQ16

Let R be an integral domain. We say that a function d:R0 → N ∪ 0 is a _________________ on R if the following conditions are satisfied.

d(a) ≤d ∀ a,b ∈R0 and

for any a,b ∈R, b ≠0 ∃ q,r∈R such that a=bq.r, where r=0 or dr<db.

*Euclidean Evaluation*

1.0000000

0.0000000

FBQ17

Let F be a field and f(x) ∈ Fx, we say that an element a∈ F is a ___________ (where) m is positive integer of f(x) if (x-a)m|f(x) but (x-a)m+1×f1

*Root of multiplicity m*

1.0000000

0.0000000

FBQ18

Let F be a field and f(x) ∈ Fx we say that an element a ∈ F is a ____________ (or zero) of f(x) if f(a) = 0

*Factor*

1.0000000

*Divides*

1.0000000

FBQ19

If S is set, an object ‘a’ in the collection S is called an_________________ of S

*Element*

1.0000000

0.0000000

FBQ20

A set with _____________element in S is called an empty set

*No*

1.0000000

0.0000000

FBQ21

____________ method is sometimes used to list the element of a large set

*Roster*

1.0000000

0.0000000

FBQ22

The set of rational numbers and the set of real numbers are respectively represented by the symbol {#1} and {#2}

10434 10435,10436

FBQ22 {100:SHORTANSWER:%100%Q}

Q

1.0000000

FBQ22 {100:SHORTANSWER:%100%R}

R

1.0000000

FBQ23

The symbol ∃ denotes __________________.

*There exist*

1.0000000

0.0000000

FBQ24

If A and B are two subsets of a set S, we can collect the element that are common to both A and B, we call this set the _______________of A and B.

*Intersection*

1.0000000

0.0000000

FBQ25

A relation R defined on a set S is said to be ________________ if we have aRa ∀ a ∈S.

*Reflexive*

1.0000000

0.0000000

0.0000000

FBQ26

A relation R defined on a set S is said to be ________________if

a R b ⇒ b R a ∀ a,b ∈ S.

*Symmetric*

1.0000000

0.0000000

FBQ27

A relation R defined on a set S is said to be ________________ if a R b and b R a ∀ a,b,c∈ S

*Transitive*

1.0000000

0.0000000

FBQ28

A relation R defined on a set S that is reflexive, symmetric and transitive is called ________________ relation

*Equivalence*

1.0000000

0.0000000

FBQ29

A ________________ f from a non – empty set A to a non – empty set B is a rule which associates with every element of A exactly on element of B

*Function*

1.0000000

0.0000000

FBQ30

A function f : A → B is called ______________ if associates different elements of A with different element of B

*Injective*

1.0000000

*One to one*

1.0000000

FBQ31

A function f : A → B is called ____________ if the range of f is B.

*Onto*

1.0000000

*Surjective*

1.0000000

FBQ32

Consider two non – empty set A and B, we define the function π1a,b=a.π1 is called the __________________ of A×B onto A

*Projection*

1.0000000

0.0000000

FBQ33

A function that is both one to one and onto is called ________________

*Bijective*

1.0000000

0.0000000

FBQ34

Any set which is equivalent to the set 1,2,…,n, for some n ∈ N, is called a __________________ set.

*Finite*

1.0000000

0.0000000

FBQ35

A set that is not ______________ is called infinite set

*Finite*

1.0000000

0.0000000

FBQ36

A function f : A → B has an inverse if and only if is __________________

*Bijective*

1.0000000

0.0000000

FBQ37

A natural number p(≠1) is called _____________ if its only divisor are 1 and p

*Prime*

1.0000000

0.0000000

FBQ38

If a natural number n(≠1) is not a prime, then it is called a _____________ number

*Composite*

1.0000000

0.0000000

FBQ39

Let A be any set, the function IA :A→A : IAa=a is called _______________ on A.

Identity function

1.0000000

0.0000000

FBQ40

Let S be a non – empty set, any function S×S → S is called a ______________ on S.

*Binary operation*

1.0000000

0.0000000

FBQ41

Let * be a binary operation on a set S. we say that: * is ____________ on a subset T of S if a*b ∈ T ∀ a,b ∈ T

*Closed*

1.0000000

0.0000000

FBQ42

Let * be a binary operation on a set S. we say that: * is ____________ if, for all a,b,c ∈ S, (a*b)*c = a ×(b*c).

*Associative*

1.0000000

0.0000000

FBQ43

Let * be a binary operation on a set S. we say that: * is ____________ if for all a,b|s, a*b = b*a

*Commutative*

1.0000000

0.0000000

FBQ44

If ° and * are two binary operations on a set S, we say that * is __________.

*Distributive over*

1.0000000

0.0000000

FBQ45

.Let * be a binary operation on a set S. if there is an element e ∈ S such that ∀ a∈ S, a * e = a and e* a = a then e is called an________________ for *.

*Identity element*

1.0000000

0.0000000

FBQ46

The Cayley table is named after the famous mathemathecian

*Arthur Cayley*

1.0000000

0.0000000

FBQ47

____________ system consists of a set with a binary operation which satisfies certain properties is called a group

*Algebraic*

1.0000000

0.0000000

FBQ48

Let G be a group, for a ∈ G, we define

a0=e

a0=an-1, if n>0

a-a=(a-1)n, if n>0

n is called the exponent ( or index) of ____________ an of a

*The integral power*

1.0000000

*integral power*

1.0000000

FBQ49

≡ is an equivalence relation, and hence partition Z into disjoint equivalence classes called ____________ modulo n.

Congruence class

1.0000000

0.0000000

FBQ50

If the set X is finite, say X = (1,2,3, …, n) then we denote S(x) by Sn and each of Sn is called a _______________ on n symbols

*Permutation*

1.0000000

0.0000000

Email: Solutions2tma@gmail.com

Whatsapp: 08155572788

## NOUN e-Exams Past Questions & Answers: MTH211 - Abstract Algebra

### Who is online

Users browsing this forum: No registered users and 2 guests