Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of unity. Choose the correct statement(s) from below:
(A) $\mathbb{C}$ is algebraic over $K$;
(B) $K$ has countably many elements;
(C) Irreducible polynomials in $K[X]$ do not have multiple roots;
(D) The characteristic of $K$ is zero.

Let $n$ be a positive integer such that every group of order $n$ is cyclic. Show the following.
(A) For all prime numbers $p$, $p^2$ does not divide $n$.
(B) If $p$ and $q$ are prime divisors of $n$, then $p$ does not divide $q - 1$. (Hint: Consider $2 \times 2$ matrices $$\left[\begin{array}{ll} x & y \\ 0 & 1 \end{array}\right]$$ with $x, y \in \mathbb{Z}/q\mathbb{Z}$ and $x^p = 1$.)
(C) Show that $(n, \phi(n)) = 1$, where $\phi(n)$ is the number of integers $m$ such that $1 \leq m \leq n$ with $\gcd(n, m) = 1$.

Let $F$ be a field and $G = \mathrm{GL}_n(F)$. For $g \in G$, write $C_g = \{hgh^{-1} \mid h \in G\}$. Let $X = \{C_g \mid g \in G,\ \text{the order of } g \text{ is } 2\}$. Determine $|X|$.

Let $f(X) \in \mathbb{Z}[X]$ be a monic polynomial. Suppose that $\alpha \in \mathbb{C}$ and $3\alpha$ are roots of $f$.
(A) Show that $f(0) \neq 1$. (Hint: if $\zeta$ and $\zeta'$ are complex numbers satisfying monic polynomials in $\mathbb{Z}[X]$, then $\zeta\zeta'$ satisfies a monic polynomial in $\mathbb{Z}[X]$.)
(B) Assume that $f$ is irreducible. Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of $f$. Let $\sigma$ be a field automorphism of $K$ such that $\sigma(\alpha) = 3\alpha$. Show that $\sigma$ has finite order and that $\alpha = 0$.

Let $G$ be a group and $N$ be a proper normal subgroup. Pick the true statement(s) from below.
(A) If $N$ and the quotient $G / N$ is finite, then $G$ is finite.
(B) If the complement $G \backslash N$ of $N$ in $G$ is finite, then $G$ is finite.
(C) If both $N$ and the quotient $G / N$ are cyclic, then $G$ is cyclic.
(D) $G$ is isomorphic to $N \times G / N$.

Let $R$ denote the ring of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, where addition and multiplication are given, respectively, by $(f + g)(x) = f(x) + g(x)$ and $(fg)(x) = f(x)g(x)$ for every $f, g \in R$ and $x \in \mathbb{R}$. A zero-divisor in $R$ is a non-zero $f \in R$ such that $fg = 0$ for some non-zero $g \in R$. Pick the true statement(s) from below:
(A) $R$ has zero-divisors.
(B) If $f$ is a zero-divisor, then $f^{2} = 0$.
(C) If $f$ is a non-constant function and $f^{-1}(0)$ contains a non-empty open set, then $f$ is a zero-divisor.
(D) $R$ is an integral domain.

Let $f(x) = x^{2} + ax + b \in \mathbb{F}_{3}[X]$. What is the number of non-isomorphic quotient rings $\mathbb{F}_{3}[X] / (f(X))$?

Let $M \in M_{n}(\mathbb{C})$. Show that $M$ is diagonalizable if and only if for every polynomial $P(X) \in \mathbb{C}[X]$ such that $P(M)$ is nilpotent, $P(M) = 0$.

Which of the following can not be the class equation for a group of appropriate order?
(A) $14 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 7$.
(B) $18 = 1 + 1 + 1 + 1 + 2 + 3 + 9$.
(C) $6 = 1 + 2 + 3$.
(D) $31 = 1 + 3 + 6 + 6 + 7 + 8$.

Let $K$ be a field of order 243 and let $F$ be a subfield of $K$ of order 3. Pick the correct statement(s) from below.
(A) There exists $\alpha \in K$ such that $K = F ( \alpha )$.
(B) The polynomial $x ^ { 242 } = 1$ has exactly 242 solutions in $K$.
(C) The polynomial $x ^ { 26 } = 1$ has exactly 26 roots in $K$.
(D) Let $f ( x ) \in F [ x ]$ be an irreducible polynomial of degree 5. Then $f ( x )$ has a root in $K$.

Let $G$ be a finite group and $X$ the set of all abelian subgroups $H$ of $G$ such that $H$ is a maximal subgroup of $G$ (under inclusion) and is not normal in $G$. Let $M$ and $N$ be distinct elements of $X$. Show the following:
(A) The subgroup of $G$ generated by $M$ and $N$ is contained in the centralizer of $M \cap N$ in $G$.
(B) $M \cap N$ is the centre of $G$.

Consider the following statement: Let $F$ be a field and $R = F [ X ]$ the polynomial ring over $F$ in one variable. Let $I _ { 1 }$ and $I _ { 2 }$ be maximal ideals of $R$ such that the fields $R / I _ { 1 } \simeq R / I _ { 2 } \neq F$. Then $I _ { 1 } = I _ { 2 }$.
Prove or find a counterexample to the following claims:
(A) The above statement holds if $F$ is a finite field.
(B) The above statement holds if $F = \mathbb { R }$.

Let $\mathrm { O } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of orthogonal matrices, i.e., matrices $A$ satisfying $A ^ { \operatorname { tr } } A = I$. Let $\mathrm { B } _ { + } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of upper triangular matrices with positive entries on the diagonal.
(A) Let $A \in \mathrm { GL } ( 2 , \mathbb { R } )$. Show that there exist $A _ { o } \in \mathrm { O } ( 2 , \mathbb { R } )$ and $A _ { b } \in \mathrm { B } _ { + } ( 2 , \mathbb { R } )$ such that $A = A _ { o } A _ { b }$. (Hint: use appropriate elementary column operations.)
(B) Show that the map $$\phi : \mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } ) \longrightarrow \mathrm { GL } ( 2 , \mathbb { R } ) \quad \left( A ^ { \prime } , A ^ { \prime \prime } \right) \mapsto A ^ { \prime } A ^ { \prime \prime }$$ is injective.
(C) Show that $\mathrm { GL } ( 2 , \mathbb { R } )$ is homeomorphic to $\mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } )$. (Hint: first show that the map $A \mapsto A _ { b }$ is continuous.)

Let $F$ be a field of characteristic $p > 0$ and $V$ a finite-dimensional $F$-vector-space. Let $\phi \in \mathrm { GL } ( V )$ be an element of order $p ^ { 3 }$. Show that there exists a basis of $V$ with respect to which $\phi$ is given by an upper-triangular matrix with 1's on the diagonal.

Let $\zeta _ { 5 } \in \mathbb { C }$ be a primitive 5th root of unity; let $\sqrt [ 5 ] { 2 }$ denote a real 5th root of 2, and let $l$ denote a square root of $-1$. Let $K = \mathbb { Q } \left( \zeta _ { 5 } , \sqrt [ 5 ] { 2 } \right)$.
(A) Find the degree $[ K : \mathbb { Q } ]$ of the field $K$ over $\mathbb { Q }$.
(B) Determine if $l \in \mathbb { Q } \left( \zeta _ { 5 } \right)$. (Hint: You may use, without proof, the following fact: if $\zeta _ { 20 } \in \mathbb { C }$ is a primitive 20th root of unity, then $\left[ \mathbb { Q } \left( \zeta _ { 20 } \right) : \mathbb { Q } \right] > 4$.)
(C) Determine if $l \in K$.

By a simple group, we mean a group $G$ in which the only normal subgroups are $\left\{ 1 _ { G } \right\}$ and $G$. Pick the correct statement(s) from below.
(A) No group of order 625 is simple.
(B) $\mathrm { GL } ( 2 , \mathbb { R } )$ is simple.
(C) Let $G$ be a simple group of order 60. Then $G$ has exactly six subgroups of order 5 .
(D) Let $G$ be a group of order 60. Then $G$ has exactly seven subgroups of order 3 .

Pick the correct statement(s) from below.
(A) There exists a finite commutative ring $R$ of cardinality 100 such that $r ^ { 2 } = r$ for all $r \in R$.
(B) There is a field $K$ such that the additive group ( $K , +$ ) is isomorphic to the multiplicative group ( $K ^ { \times } , \cdot$ ).
(C) An irreducible polynomial in $\mathbb { Q } [ x ]$ is irreducible in $\mathbb { Z } [ x ]$.
(D) A monic polynomial of degree $n$ over a commutative ring $R$ has at most $n$ roots in $R$.

Let $G$ be a finite group that has a non-trivial subgroup $N$ (i.e. $\left\{ 1 _ { G } \right\} \neq N \neq G$ ) that is contained in every non-trivial subgroup of $G$. Show that
(A) $G$ is a $p$-group for some prime number $p$;
(B) $N$ is a normal subgroup of $G$.

Let $F$ be a field such that it has a finite non-Galois extension field. Let $V$ be a finite-dimensional vector-space over $F$. Let $V _ { 1 } , \ldots , V _ { r }$ be proper subspaces of $V$. Prove or disprove the following assertion: $V \neq \bigcup _ { i = 1 } ^ { r } V _ { i }$.

For a ring homomorphism $R \longrightarrow S$ (of commutative rings) and an ideal $I$ of $R$, the fibre over $I$ is the ring $S / I S$, i.e., the quotient of $S$ by the $S$-ideal generated by the image of $I$ in $S$. Let $S = \mathbb { C } [ X , Y ] / ( X Y - 1 )$ and $R = \mathbb { C } [ x + \alpha y ]$ where $\alpha \in \mathbb { C }$ and $x , y$ are the images of $X , Y$ in $S$. Consider the ring homomorphism $R \subseteq S$. Let $I = ( x + \alpha y - \beta ) R$, where $\beta \in \mathbb { C }$. For each nonnegative integer $n$, determine the set of ( $\alpha , \beta$ ) such that the fibre over $I$ has exactly $n$ maximal ideals.

Which of the following groups are cyclic?
(A) $\mathbb { Z } / 2 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(B) $\mathbb { Z } / 3 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(C) Every group of order 18.
(D) $\left( \mathbb { Q } ^ { \times } , \cdot \right)$

(A) (3 marks) Let $G$ be a group such that $| G | = p ^ { a } d$ with $a \geq 1$ and $( p , d ) = 1$. Let $P$ be a Sylow $p$-subgroup and let $Q$ be any $p$-subgroup of $G$ such that $Q$ is not a subgroup of $P$. Show that $P Q$ is not a subgroup of $G$.
(B) (7 marks) Let $\Gamma$ be a group that is the direct product of its Sylow subgroups. Show that every subgroup of $\Gamma$ also satisfies the same property.

Let $U ( n )$ be the group of $n \times n$ unitary complex matrices. Let $P \subset U ( n )$ be the set of all finite order elements of $U ( n )$, that is, $P = \left\{ X \in U ( n ) \mid X ^ { m } = 1 \text{ for some } m \geq 1 \right\}$. Show that $P$ is dense in $U ( n )$.

Let $G$ (respectively, $H$ ) be a Sylow 2-subgroup (respectively, Sylow 7-subgroup) of the symmetric group $S _ { 17 }$. Pick the correct statement(s) from below.
(A) The order of $G$ is $2 ^ { 15 }$.
(B) $H$ is abelian.
(C) $G$ has a subgroup isomorphic to $\mathbb { Z } / 8 \mathbb { Z } \times \mathbb { Z } / 8 \mathbb { Z }$.
(D) If $\sigma \in S _ { 17 }$ has order 4 , then $\sigma$ is a 4-cycle.

Let $k$ be a finite field of characteristic $p > 2$ and $G$ the subgroup of $\mathrm { GL } _ { 2 } ( k )$ consisting of all matrices whose first column is $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$. Pick the correct statement(s) from below.
(A) $G$ is a normal subgroup of $\mathrm { GL } _ { 2 } ( k )$.
(B) $G$ is a $p$-group.
(C) $\left\{ \left[ \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right] : a \in k \right\}$ is a normal subgroup of $G$.
(D) $G$ is abelian.