2000

1) Let n be a fixed positive integer and let Zn be the ring of integers modulo n. Let G={ˉa∈Zn∣ˉa≠¯0 and a is relatively prime to n} . Show that G is a group under multiplication defined in Zn. Hence or otherwise, show that aϕ(n)≡a (mod n) for all integers a relatively prime to n where ϕ (n) denotes the number of positive integers that are less than n and are relatively prime to n.

[12M]

2) Let M be a subgroup and N a normal subgroup of a group G. Show that MN is a subgroup of G and MN/N is isomorphic to M/(M ∩N).

[12M]

3) Let F be a finite field. Show that the characteristic of must be a prime integer p and the number of elements in F must be pm for some positive integer m.

[20M]

4) Let F be a field and F[x] denote the set of all polynomials defined over F. If f(x) is an irreducible polynomial in F(x), show that the ideal generated by f(x) in F(x) is maximal and F[x]/(f(x)) is a field.

[20M]

4) Show that any finite commutative ring with no zero divisors must be a field.

[20M]

1999

1) If ϕ is a homomorphism of G into G with kernel K,then show that K, is a normal subgroup of G.

[20M]

2) If p is a prime number and pα|o(G),then prove that G has a subgroup of of order pα

[20M]

3) Let R be a commutative ring with unit element where only ideal are (o) and R itself.Show that R is a field.

[20M]

4) Let A be a subset of the metric space (M,ρ)..If (A,ρ) is compact, then show that A is closed subset of (M,ρ).

[20M]

5) A sequence {SR} is defined by the recursion formula Sn+1=√3Sn,S1=1.Does this sequence converge ?If so,find limSn.

[20M]

TBC

6) Test for convergence the integral ∫10xp(log1x)pdx.

[20M]

1998

1) Prove that if a group has only four elements then it must be abelian.

2) If H and K are subgroups of a group G, then show that HK is subgroup of G if and only if HK=KH.

3) Show that every group of order 15 has a normal subgroup of order 5.

4) Let (R,+,.) be a system satisfying all the axioms for a ring with unity with the possible exception of a+b=b+a. Prove that (R,+,.) is a ring.

5) If p is prime then prove that Zp is a field. Discuss the case when p is not a prime number.

6) Let D be a principal ideal domain. Show that every element that is neither zero nor a unit in D is a product of irreducibles.

7) Let X ba a metric space and E⊂X.Show that interior of E is the largest open set contained in E, boundary of E=(closure of E)∩(closure of X-E).

1997

1) Show that a necessary and sufficient condition for a subset Hof a group G to be a subgroup is HH−1=H

[10M]

2) Show that the order of each subgroup of a finite group is a divisor of the order of the group.

[10M]

3) In a group G, the commutator (a,b),a,b∈G is the element aba−1b−1 and the smallest subgroup containing all commutators is called the commutator subgroup of G. Show that a quotient group G/H is abelian if and only if H contains the commutator subgroup of G.

[10M]

4) If x2=x for all x in a ring R, show that R is commutative. Give an example to show that the converse is not true.

[10M]

5) Show that an ideal S of the ring of integers Z is a maximal ideal if and only if S is generated by a prime integer.

[10M]

6) Show that in an integral domain every prime element is irreducible. Give an example to show that the converse is not true.

[10M]

1996

1) Let R be the set of real numbers and G={(a,b)∣a,b∈R,a≠0} ∗G×G→G is defined by (a,b)∗(c,d)=(ac,be+d).
Show that (G,∗) is a group. Is it abelian?
Is (H,∗) a subgroup of (G,∗), when H={(1,b)∣b∈R}?

[12M]

2) Let f be a homomorphism of a group G onto a group G’ with kernel H. For each subgroup K′ of G’ define K by K={x∈G∣f(x)∈K′} Prove that (i) K′ is isomorphic to K/H. (ii) G/K is isomorphic to G′/K′

[12M]

3) Prove that a normal subgroup H of a group G is maximal, if and only if the quotient group G/H is simple.

[12M]

4) In a ring R, prove that cancellation laws hold, if and only if R has no zero divisors.

[15M]

5) If S is an ideal of a ring R and T any subring or R, then prove that S is an ideal of S+T={s+t∣s∈S,t∈T}

[15M]

6) Prove that the polynominl x2+x+4 is irreducible over the field of integers modulo 11 .

[15M]

1995

1) Let G be a finite set closed under an associative binary operation such that ab=ac implies b=c and ba=ca implies b=c for all a,b,c∈G. Prove that G is a group.

[12M]

2) Let G be a group of order pn, where p is a prime number and n>0. Let H be a proper subgroup of G and N(H)={x∈G:x−1hx∈H∀h∈H}. Prove that N(H)≠H.

[12M]

3) Show that a group of order 112 is not simple.

[12M]

4) Let R be a ring with identity. Suppose there is an element a of R which has more than one right inyerse Prove that a has infinitely many right inverses

[15M]

5) Let F be a field and let p(x) be an irreducible polynomial over F. Let <p(x)> be the ideal generated by p(x). Prove that <p(x)> is a maximal ideal.

[15M]

6) Let F be a field of characteristic p≠0. Let F(x) be the polynomial ring. Suppose f(x)=a0+a1x+…+anxn is an element of F[x]. Define f(x)=a1+2a2x+…+nanxn−1 If f′(x)=0, prove that there exists g(x)∈F[x] such that f(x)=g(xp).

[15M]