Let $R$ be an integral domain such that every non-zero prime ideal of $R[X]$ (where $X$ is an indeterminate) is maximal. Choose the correct statement(s):
(A) $R$ is a field;
(B) $R$ contains $\mathbb{Z}$ as a subring;
(C) Every ideal in $R[X]$ is principal;
(D) $R$ contains $\mathbb{F}_{p}$ as a subring for some prime number $p$.

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be such that $\int_{-\infty}^{\infty}|f(x)|\,\mathrm{d}x < \infty$. Define $F : \mathbb{R} \longrightarrow \mathbb{R}$ by $F(x) = \int_{-\infty}^{x} f(t)\,\mathrm{d}t$. Choose the correct statement(s):
(A) $f$ is continuous;
(B) $F$ is continuous;
(C) $F$ is uniformly continuous;
(D) There exists a positive real number $M$ such that $|f(x)| < M$ for all $x \in \mathbb{R}$.

Let $\omega \in \mathbb{C}$ be a primitive third root of unity. How many distinct possible images of $\omega$ are there under all the field homomorphisms $\mathbb{Q}(\omega) \longrightarrow \mathbb{C}$.

Let $C := \{z \in \mathbb{C} : |z| = 5\}$. What is value of $M$ such that $$2\pi\imath M = \int_{C} \frac{1}{z^{2}-5z+6}\,\mathrm{d}z\,?$$

Consider the set $\mathbb{R}[X]$ of polynomials in $X$ with real coefficients as a real vector space. Let $T$ be the $\mathbb{R}$-linear operator on $\mathbb{R}[X]$ given by $$T(f) = \frac{\mathrm{d}^{2}f}{\mathrm{d}X^{2}} - \frac{\mathrm{d}f}{\mathrm{d}X} + f$$ What is the nullity of $T$?

Let $f \in \mathbb{R}[x,y]$ be such that there exists a non-empty open set $U \subseteq \mathbb{R}^{2}$ such that $f(x,y) = 0$ for every $(x,y) \in U$. Show that $f = 0$.

Let $A \in M_{n \times n}(\mathbb{C})$.
(a) Suppose that $A^{2} = 0$. Show that $\lambda$ is an eigenvalue of $(I_{n}+A)$ if and only if $\lambda = 1$. ($I_{n}$ is the $n \times n$ identity matrix.)
(b) Suppose that $A^{2} = -1$. Determine (with proof) whether $A$ is diagonalizable.

Let $f$ be a non-constant entire function satisfying the following conditions:
(a) $f(0) = 0$;
(b) For every positive real number $M$, the set $\{z : |f(z)| < M\}$ is connected.
Prove that $f(z) = cz^{n}$ for some constant $c$ and positive integer $n$.

Let $\left(a_{mn}\right)_{m \geq 1, n \geq 1}$ be a double sequence of real numbers such that
(a) For every $n$, $b_{n} := \lim_{m \rightarrow \infty} a_{mn}$ exists;
(b) For all strictly increasing sequences $\left(m_{k}\right)_{k \geq 1}$ and $\left(n_{k}\right)_{k \geq 1}$ of positive integers, $\lim_{k \rightarrow \infty} a_{m_{k}n_{k}} = 1$.
Show that the sequence $\left(b_{n}\right)_{n \geq 1}$ converges to $1$.

Let $f \in \mathbb{C}[x,y]$ be such that $f(x,y) = f(y,x)$. Show that there is a $g \in \mathbb{C}[x,y]$ such that $f(x,y) = g(x+y, xy)$.

Let $X$ be a topological space and $f : X \longrightarrow [0,1]$ be a closed continuous surjective map such that $f^{-1}(a)$ is compact for every $0 \leq a \leq 1$. Prove or disprove: $X$ is compact. (A map is said to be closed if it takes closed sets to closed sets.)

Let $U = \left\{ (x, y) \in \mathbb{R}^2 \mid 1 < x^2 + y^2 < 4 \right\}$. Let $p, q \in U$. Show that there is a continuous map $\gamma : [0,1] \longrightarrow U$ such that $\gamma(0) = p$ and $\gamma(1) = q$ and such that $\gamma$ is differentiable on $(0,1)$.

If $I, J$ are two maximal ideals in a PID that is not a field, then show that $IJ$ is never a prime ideal.

Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function. Suppose that $f(z) \in \mathbb{R}$ if $z$ is on the real axis or on the imaginary axis. Show that $f'(z) = 0$ at $z = 0$.

Let $A \subseteq \mathbb{R}^n$ be a closed proper subset. For $x, y \in \mathbb{R}^n$, denote the usual (Euclidean) distance between them by $d(x, y)$. Let $x \in \mathbb{R}^n \setminus A$; define $\delta := \inf\{ d(x, y) \mid y \in A \}$. Show that there exists $y \in A$ such that $\delta = d(x, y)$.

Let $F$ be a field and $V$ a finite-dimensional vector-space over $F$. Let $T : V \longrightarrow V$ be a linear transformation, such that for every $v \in V$, there exists $n \in \mathbb{N}$ such that $T^n(v) = v$.
(A) Show that if $F = \mathbb{C}$, then $T$ is diagonalizable.
(B) Show that if $\operatorname{char}(F) > 0$, then there exists a non-diagonalizable $T$ satisfying the above hypothesis.

Let $F = \mathbb{Q}(\omega, \sqrt[3]{2})$, where $\omega \in \mathbb{C}$ is a primitive cube-root of unity. Find a $\mathbb{Q}$-basis for $F$ (with proof). Let $\mu : F \longrightarrow F$ be the $\mathbb{Q}$-linear map given by $\mu(a) = \omega^2 a$. Find the matrix of $\mu$ with respect to the basis obtained above.

Let $G$ be a non-trivial subgroup of the group $(\mathbb{R}, +)$. Show that either $G$ is dense in $\mathbb{R}$ or that $G = \mathbb{Z} \cdot l$ where $l = \inf\{ x \in G \mid x > 0 \}$.

Let $G$ be a subgroup of the group of permutations on a finite set $X$. Let $F$ be the $\mathbb{C}$-vector-space of all the functions from $X$ to $\mathbb{C}$. $G$ acts on $F$ by $(g \cdot f) : x \mapsto f(g^{-1}(x))$. Show that there is a $\phi \in F$ such that $g \cdot \phi = \phi$ for every $g \in G$. Show that there is a subspace $F'$ of $F$ such that $F = F' \oplus \mathbb{C}\langle \phi \rangle$ and such that $g \cdot f \in F'$ for every $g \in G$ and $f \in F'$.

(A) Let $A$ and $B$ be $n \times n$ matrices with entries in $\mathbb{N}$. Show that if $B = A^{-1}$ then $A$ and $B$ are permutation matrices. (A permutation matrix is a matrix obtained by permuting the rows of the identity matrix.)
(B) Let $A$ be an $n \times n$ complex matrix that is not a scalar multiple of $I_n$. Show that $A$ is similar to a matrix $B$ such that $B_{1,1}$ (i.e. the top left entry of $B$) is 0.

Let $S^1 = \{ z \in \mathbb{C} : |z| = 1 \}$. Consider the map $\mathrm{Sq} : S^1 \longrightarrow S^1$, $$\operatorname{Sq}(z) = z^2$$ Show that there does not exist a continuous map $\mathrm{Sqrt} : S^1 \longrightarrow S^1$ such that $\mathrm{Sq} \circ \mathrm{Sqrt} = Id_{S^1}$. (That is, $(\operatorname{Sqrt}(w))^2 = w$.) (Hint: If such a map existed, show that there would be a bijective continuous map $S^1 \times \{1, -1\} \longrightarrow S^1$.)

Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ (where $\mathbb{R}$ is the set of all real numbers) that satisfies the following property: For every natural number $n$ $$f(n) = \text{the smallest prime factor of } n.$$ For example, $f(12) = 2$, $f(105) = 3$. Calculate the following.
(a) $\lim_{x \rightarrow \infty} f(x)$.
(b) The number of solutions to the equation $f(x) = 2016$.

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.

Consider the ideal $I := (ux, uy, vx, uv)$ in the polynomial ring $\mathbb{Q}[u,v,x,y]$, where $u,v,x,y$ are indeterminates. Choose the correct statement(s) from below:
(A) Every prime ideal containing $I$ contains the ideal $(x,y)$;
(B) Every prime ideal containing $I$ contains the ideal $(x,y)$ or the ideal $(u,v)$;
(C) Every maximal ideal containing $I$ contains the ideal $(u,v)$;
(D) Every maximal ideal containing $I$ contains the ideal $(u,v,x,y)$.

Let $f$ be an irreducible cubic polynomial over $\mathbb{Q}$ with at most one real root and $\mathbb{k}$ the smallest subfield of $\mathbb{C}$ containing the roots of $f$. Choose the correct statement(s) from below:
(A) $\sigma(K) \subseteq K$ where $\sigma$ denotes complex conjugation;
(B) $[K : \mathbb{Q}]$ is an even number;
(C) $[(K \cap \mathbb{R}) : \mathbb{Q}]$ is an even number;
(D) $K$ is uncountable.