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Q. Find the dimension of the subspace of R4 that is span of the vectors

(█(1¦(-1)@0@1)), (█(2¦1@1@1)),(█(0¦0@0@0)),(█(1¦1@-2@-5))

Q. Choose the correct answer.

Q. Let b and c are elements in a group G and e is identity element of G. If b5=c3=e,then inverse of bcb2 is

a. b2cb

b. b3c2b4

d. b2c2b4

(█(1¦(-1)@0@1)), (█(2¦1@1@1)),(█(0¦0@0@0)),(█(1¦1@-2@-5))

Q. Choose the correct answer.

Q. Let b and c are elements in a group G and e is identity element of G. If b5=c3=e,then inverse of bcb2 is

a. b2cb

b. b3c2b4

d. b2c2b4

Let N ∈d , where d ≠1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d].

1. N(x) = 0 if x=0.

2 N(xy) = N(x)N(y)

3. N(x) =1 if x is a unit

4. N(x) is prime if x is irreducible.

1. N(x) = 0 if x=0.

2 N(xy) = N(x)N(y)

3. N(x) =1 if x is a unit

4. N(x) is prime if x is irreducible.

A linear equation over R can have at most one root in C\R.

Is the statement true or false? Justify your answer.

Is the statement true or false? Justify your answer.

a) Let d∈N , where d ≠1 and d is not divisible by the square of a prime.

Prove that N:Z[√d]→N∪{0}:N(a+b√d)=|a²-db²| satisfies the following properties for x,y∈Z[√d] :

i) N(x) = 0⇔x=0

ii) N(xy) = N(x)N(y)

iii) N(x) =1⇔ x is a unit

iv) N(x) is prime ⇒x is irreducible in Z[√d ] .

b) Prove or disprove that C≃ R as fields.

Prove that N:Z[√d]→N∪{0}:N(a+b√d)=|a²-db²| satisfies the following properties for x,y∈Z[√d] :

i) N(x) = 0⇔x=0

ii) N(xy) = N(x)N(y)

iii) N(x) =1⇔ x is a unit

iv) N(x) is prime ⇒x is irreducible in Z[√d ] .

b) Prove or disprove that C≃ R as fields.

Prove that N:Z[√d]→N∪{0}:N(a+b√d)=|a²-db²| satisfies the following properties for x,y∈Z[√d] :

i) N(x) = 0⇔x=0

ii) N(xy) = N(x)N(y)

iii) N(x) =1⇔ x is a unit

iv) N(x) is prime ⇒x is irreducible in Z[√d ] .

b) Prove or disprove that C≃ R as fields.

a) Give two distinct maximal ideals of R[x] , with justification.

b) Give an example, with justification, of rings R and S with a ring homomorphism

φ:R→S such that P is a prime ideal of R but φ(P) is not a prime ideal of S.

c) Let D be a Euclidean domain, with Euclidean valuation d . Prove that if n∈N such

that d(1) + n≥0 , then f :D \ {0}→Z : f (x) = d(x) + n is also a Euclidean valuation

on D .

b) Give an example, with justification, of rings R and S with a ring homomorphism

φ:R→S such that P is a prime ideal of R but φ(P) is not a prime ideal of S.

c) Let D be a Euclidean domain, with Euclidean valuation d . Prove that if n∈N such

that d(1) + n≥0 , then f :D \ {0}→Z : f (x) = d(x) + n is also a Euclidean valuation

on D .

a) Consider the ring R= Z(subscript)3[x] / < x^8 - 1>

i) Is R a finite ring?

ii) Does R have zero divisors?

iii) Does R have nilpotent elements?

Give reasons for your answers.

b) Give an example, with justification, of a subset of a ring that is a subgroup under

addition but not a subring.

c) Construct a multiplication (Cayley) table for 3Z/9Z.

i) Is R a finite ring?

ii) Does R have zero divisors?

iii) Does R have nilpotent elements?

Give reasons for your answers.

b) Give an example, with justification, of a subset of a ring that is a subgroup under

addition but not a subring.

c) Construct a multiplication (Cayley) table for 3Z/9Z.

a) If G is a group of order 40, and H and K are its subgroups of orders 20 and 10,

then check whether or not HK ≤ G . Further, show that o(H∩K) ≥ 5.

b) Prove that C*/S~=R+, where S ={z∈C*| |z|=1}, R+ ={x∈R| x>0} and C*=C\{0}.

c) What are the possible algebraic structures of a group of order 99?

then check whether or not HK ≤ G . Further, show that o(H∩K) ≥ 5.

b) Prove that C*/S~=R+, where S ={z∈C*| |z|=1}, R+ ={x∈R| x>0} and C*=C\{0}.

c) What are the possible algebraic structures of a group of order 99?

a)Let S be a set with n elements, n >= 3 . Let B be the set of bijective mappings of S

onto itself.

i) Check whether (B, o) is a group or not.

ii) Give the cardinality of the set B .

iii) Is o commutative? Give reasons for your answer.

b) Check whether or not H = {x∈R*|x =1 or x∉Q } is a subgroup of ( R*,.) . Also

check whether K = {x∈R*|x>=1} is a subgroup of R* or not.

c) Let U(n) ={x∈N|1<+x<n,(x,n)=1}. Show that U(n) is a group w.r.t.

multiplication modulo n . Also show that U(14) is cyclic and U(20) is not

cyclic.

d) Obtain the centre of Q_8 and two distinct right cosets of Z(Q_8 ) in Q_8.

onto itself.

i) Check whether (B, o) is a group or not.

ii) Give the cardinality of the set B .

iii) Is o commutative? Give reasons for your answer.

b) Check whether or not H = {x∈R*|x =1 or x∉Q } is a subgroup of ( R*,.) . Also

check whether K = {x∈R*|x>=1} is a subgroup of R* or not.

c) Let U(n) ={x∈N|1<+x<n,(x,n)=1}. Show that U(n) is a group w.r.t.

multiplication modulo n . Also show that U(14) is cyclic and U(20) is not

cyclic.

d) Obtain the centre of Q_8 and two distinct right cosets of Z(Q_8 ) in Q_8.

1. Which of the following statements are true? Give reasons for your answers.

i) If f : S->S is 1-1, where S is a set, then f is onto.

ii) The signature of the product of k disjoint cycles in S_n is k .

iii) R[x] has no maximal ideal.

iv) If m| n in Z , then mZ is an ideal of Z/ nZ .

v) Every non-trivial subgroup of an infinite group is infinite.

vi) The characteristic of the quotient field of any UFD is zero.

vii) If H and K are subgroups of a group G , then HK is also a subgroup of G .

viii) If the order of a group is even, then it cannot have an element of odd order.

ix) If R is an integral domain, then R / I has no zero divisors, where I is an ideal of R .

i) If f : S->S is 1-1, where S is a set, then f is onto.

ii) The signature of the product of k disjoint cycles in S_n is k .

iii) R[x] has no maximal ideal.

iv) If m| n in Z , then mZ is an ideal of Z/ nZ .

v) Every non-trivial subgroup of an infinite group is infinite.

vi) The characteristic of the quotient field of any UFD is zero.

vii) If H and K are subgroups of a group G , then HK is also a subgroup of G .

viii) If the order of a group is even, then it cannot have an element of odd order.

ix) If R is an integral domain, then R / I has no zero divisors, where I is an ideal of R .