2000

1) Let $n$ be a fixed positive integer and let $Z_n$ be the ring of integers modulo $n$. Let $\mathrm{G}=\{\overline{a}\in Z_n\mid \overline{a}\ne \overline{0}$ and a is relatively prime to $n\}$ . Show that $\mathrm{G}$ is a group under multiplication defined in ${\mathrm{Z}}_{\mathrm{n}}$. Hence or otherwise, show that $a^{\varphi (n)}\equiv a$ (mod $n$) for all integers a relatively prime to $n$ where $\varphi$ (n) denotes the number of positive integers that are less than $\mathrm{n}$ 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 $\cap$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 $\mathrm{F}$ must be $p^m$ for some positive integer $m$.

[20M]

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

[20M]

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

[20M]

1999

1) If $\varphi$ 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 $p^{\alpha }|o(G)$,then prove that G has a subgroup of of order $p^{\alpha }$

[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 $(M,\rho ).$.If $(A,\rho )$ is compact, then show that A is closed subset of $(M,\rho )$.

[20M]

5) A sequence $\{S_R\}$ is defined by the recursion formula $S_{n+1}=\sqrt{3S_n,S_1}=1$.Does this sequence converge ?If so,find $lim{S}_n$.

[20M]

TBC

6) Test for convergence the integral ${\int }_0^1{x}^p(log{\displaystyle \frac{1}{x}}{)}^{p}dx.$

[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 $Z_p$ 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$\subset$X.Show that interior of $E$ is the largest open set contained in $E$, boundary of E=(closure of E)$\cap$(closure of X-E).

1997

1) Show that a necessary and sufficient condition for a subset Hof a group $\mathrm{G}$ to be a subgroup is ${\mathrm{H}\mathrm{H}}^{-1}=\mathrm{H}$

[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 $\mathrm{G}$, the commutator $(\mathrm{a},\mathrm{b}),\mathrm{a},\mathrm{b}\in \mathrm{G}$ is the element ${\mathrm{a}\mathrm{b}\mathrm{a}}^{-1}{\mathrm{b}}^{-1}$ 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 $\mathrm{H}$ contains the commutator subgroup of $\mathrm{G}$.

[10M]

4) If $x^2=x$ 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 $\mathrm{S}$ of the ring of integers $\mathrm{Z}$ is a maximal ideal if and only if $\mathrm{S}$ 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 $\mathrm{G}=\{(\mathrm{a},\mathrm{b})\mid \mathrm{a},\mathrm{b}\in \mathrm{R},\mathrm{a}\ne 0\}$ ${}^{\ast }\mathrm{G}\times \mathrm{G}\to \mathrm{G}$ is defined by $(a,b)\ast (c,d)=(ac,be+d)$.
Show that $(\mathrm{G},{}^{\ast })$ is a group. Is it abelian?
Is $(\mathrm{H},{}^{\ast })$ a subgroup of $(\mathrm{G},{}^{\ast }),$ when $\mathrm{H}=\{(1,\mathrm{b})\mid \mathrm{b}\in \mathrm{R}\}?$

[12M]

2) Let $f$ be a homomorphism of a group G onto a group G’ with kernel H. For each subgroup ${\mathrm{K}}^{\prime }$ of G’ define $\mathrm{K}$ by $\mathrm{K}=\{\mathrm{x}\in \mathrm{G}\mid \mathrm{f}(\mathrm{x})\in {\mathrm{K}}^{\mathrm{\prime }}\}$ Prove that (i) ${\mathrm{K}}^{\prime }$ is isomorphic to $\mathrm{K}/\mathrm{H}$. (ii) $\mathrm{G}/\mathrm{K}$ is isomorphic to ${\mathrm{G}}^{\prime }/{\mathrm{K}}^{\prime }$

[12M]

3) Prove that a normal subgroup $\mathrm{H}$ of a group $\mathrm{G}$ is maximal, if and only if the quotient group $\mathrm{G}/\mathrm{H}$ 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 $\mathrm{S}$ is an ideal of a ring $\mathrm{R}$ and $\mathrm{T}$ any subring or $\mathrm{R},$ then prove that $\mathrm{S}$ is an ideal of $\mathrm{S}+\mathrm{T}=\{\mathrm{s}+\mathrm{t}\mid \mathrm{s}\in \mathrm{S},\mathrm{t}\in \mathrm{T}\}$

[15M]

6) Prove that the polynominl $x^2+x+4$ 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 $\begin{array}{l}ab=ac\text{ implies }b=c\\ \text{ and }ba=ca\text{ implies }b=c\end{array}$ for all $a,b,c\in G.$ Prove that $G$ is a group.

[12M]

2) Let $G$ be a group of order $p^n,$ where $p$ is a prime number and $n>0.$ Let $H$ be a proper subgroup of $G$ and $N(H)=\{x\in G:x^{-1}hx\in H\mathrm{\forall }h\in H\}.$ Prove that $N(H)\ne H$.

[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 $p\ne 0.$ Let $F(x)$ be the polynomial ring. Suppose $f(x)=a_0+a_{1}x+\dots +a_n{x}^n$ is an element of $F[x]$. Define $f(x)=a_1+2a_{2}x+\dots +n{a}_n{x}^{n-1}$ If $f^{\mathrm{\prime }}(x)=0,$ prove that there exists $g(x)\in F[x]$ such that $f(x)=g\left(x^p\right)$.

[15M]