There is a field with 10 elements.

There are at least three non-isomorphic rings with 4 elements.

The group $( \mathbb { Q } , + )$ is a finitely generated abelian group.

$\mathbb { Q } ( \sqrt { 7 } )$ and $\mathbb { Q } ( \sqrt { 17 } )$ are isomorphic as fields.

There is a field with 121 elements.

There are no infinite group with subgroups of index 5.

Every finite group of odd order is isomorphic to a subgroup of $A _ { n }$, the group of all even permutations.

Every group of order 6 abelian.

Two abelian groups of the same order are isomorphic.

Let $G$ be a group. The following statements hold.
(a) If $G$ has nontrivial centre $C$, then $G / C$ has trivial centre.
(b) If $G \neq 1$, there exists a nontrivial homomorphism $h : \mathbb { Z } \rightarrow G$.
(c) If $| G | = p ^ { 3 }$, for $p$ a prime, then $G$ is abelian.
(d) If $G$ is nonabelian, then it has a nontrivial automorphism.

Let $h : \mathbb { C } \rightarrow \mathbb { C }$ be an analytic function such that $h ( 0 ) = 0 ; h \left( \frac { 1 } { 2 } \right) = 5$, and $| h ( z ) | < 10$ for $| z | < 1$. Then,
(a) the set $\{ z : | h ( z ) | = 5 \}$ is unbounded by the Maximum Principle;
(b) the set $\left\{ z : \left| h ^ { \prime } ( z ) \right| = 5 \right\}$ is a circle of strictly positive radius;
(c) $h ( 1 ) = 10$;
(d) regardless of what $h ^ { \prime }$ is, $h ^ { \prime \prime } \equiv 0$.

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb{k})$ where $\mathbb{k}$ is an algebraically closed field. Choose the correct statement(s) from below:
(A) Every element of $G$ is diagonalizable;
(B) Every element of $G$ is diagonalizable if $\mathbb{k}$ is an algebraic closure of $\mathbb{Q}$;
(C) Every element of $G$ is diagonalizable if $\mathbb{k}$ is an algebraic closure of $\mathbb{F}_p$;
(D) There exists a basis of $\mathbb{k}^n$ with respect to which every element of $G$ is a diagonal matrix.

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 $N \times G / N$.

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 $k$ 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.

Let $M _ { n } ( \mathbb { R } )$ be the space of $n \times n$ real matrices. View $M _ { n } ( \mathbb { R } )$ as a metric space with $$d \left( \left[ a _ { i , j } \right] , \left[ b _ { i , j } \right] \right) : = \max _ { i , j } \left| a _ { i , j } - b _ { i , j } \right|$$ Let $U \subset M _ { n } ( \mathbb { R } )$ be the subset of matrices $M \in M _ { n } ( \mathbb { R } )$ such that $\left( M - I _ { n } \right) ^ { n } = 0$.
(A) $U$ is closed.
(B) $U$ is open.
(C) $U$ is compact.
(D) $U$ is neither closed or open.

Is the decomposition $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ obtained unique?

We consider two matrices of $M _ { 2 } ( \mathbf { Z } )$ that are $\mathbf { C }$-equivalent. Are they always $\mathbf { Z }$-equivalent?