Calculate the minimal polynomial of $\sqrt { 2 } e ^ { \frac { 2 \pi i } { 3 } }$ over $\mathbb { Q }$.

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite-dimensional vector space over $F$, then $R$ is a field.

Let $C [ 0,1 ]$ be the space of continuous real-valued functions on the interval $[ 0,1 ]$. This is a ring under point-wise addition and multiplication. The following are true.
(a) For any $x \in [ 0,1 ]$, the ideal $M ( x ) = \{ f \in C [ 0,1 ] \mid f ( x ) = 0 \}$ is maximal.
(b) $C [ 0,1 ]$ is an integral domain.
(c) The group of units of $C [ 0,1 ]$ is cyclic.
(d) The linear functions form a vector-space basis of $C [ 0,1 ]$ over $\mathbb { R }$.

Let $F$ be a field with 256 elements, and $f \in F [ x ]$ a polynomial with all its roots in $F$. Then,
(a) $f \neq x ^ { 15 } - 1$;
(b) $f \neq x ^ { 63 } - 1$;
(c) $f \neq x ^ { 2 } + x + 1$;
(d) if $f$ has no multiple roots, then $f$ is a factor of $x ^ { 256 } - x$.

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 $f(X) \in \mathbb{Z}[X]$ 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 $R$ 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 $K$ 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 $K$.

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 $\zeta _ { 5 } \in \mathbb { C }$ 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 $l \in K$.

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.

Pick the correct statement(s) from below.
(A) There exists a maximal ideal $M$ of $\mathbb { Z } [ x ]$ such that $M \cap \mathbb { Z } = ( 0 )$.
(B) If $M$ is a maximal ideal of $\mathbb { Z } [ x ]$, then $\mathbb { Z } [ x ] / M$ is finite.
(C) If $I$ is an ideal of $\mathbb { Z } [ x ]$ such that $\mathbb { Z } [ x ] / I$ is finite, then $I$ is maximal.
(D) The ideal $\left( 7 , x ^ { 2 } - 14 x - 2 \right)$ in $\mathbb { Z } [ x ]$ is maximal.

Consider the ring $\mathcal { C } ( \mathbb { R } )$ of continuous real-valued functions on $\mathbb { R }$, with pointwise addition and multiplication. For $A \subset \mathbb { R }$, the ideal of $A$ is $I ( A ) = \{ f \in \mathcal { C } ( \mathbb { R } ) \mid f ( a ) = 0$ for all $a \in A \}$. For a subset $I$ of $\mathcal { C } ( \mathbb { R } )$, the zero-set of $I$ is $Z ( I ) = \{ a \in \mathbb { R } \mid f ( a ) = 0$ for all $f \in I \}$. Prove the following:
(A) (3 marks) $Z ( I \cap J ) = Z ( I J )$ for ideals $I$ and $J$ of $\mathcal { C } ( \mathbb { R } )$.
(B) (2 marks) For each $a \in \mathbb { R } , I ( a )$ is a maximal ideal.
(C) (3 marks) The set $\{ f \in \mathcal { C } ( \mathbb { R } ) \mid f$ has compact support $\}$ is a proper ideal, and its zero set is empty.
(D) (2 marks) True/False: For each prime ideal $\mathfrak { p }$ of $\mathcal { C } ( \mathbb { R } ) , Z ( \mathfrak { p } )$ is a singleton set. (Justify your answer.)

Let $\mathbb { F } _ { q }$ be the finite field with $q$ elements and $P \in \mathbb { F } _ { q } [ x ]$ be a monic irreducible polynomial of even degree $2 d$. Then show that $P$, when considered as a polynomial in $\mathbb { F } _ { q ^ { 2 } } [ x ]$, decomposes into a product $P = Q _ { 1 } Q _ { 2 }$ of irreducible polynomials $Q _ { i }$ in $\mathbb { F } _ { q ^ { 2 } } [ x ]$ with $\operatorname { deg } \left( Q _ { i } \right) = d$.

What can be said about $(\mathbb{A}, +, *)$?

Let $A$ be an algebraic $\mathbb{R}$-algebra without zero divisors. a) Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. b) Show that if $x \in A \setminus \mathbb{R}$, then $\mathbb{R} + \mathbb{R}x$ is an $\mathbb{R}$-algebra isomorphic to $\mathbb{C}$.

We assume that $A$ is not isomorphic to one of the algebras $\mathbb{R}$ or $\mathbb{C}$. Show that there exists $i_A \in A$ such that $i_A^2 = -1$.

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Conclude that $A$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (Theorem C).

Show that if $n$ is a strictly positive integer, the ring $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ has property (TF), but not property (F).

Show that the ring $\mathbf { Q }$ of rational numbers does not have property (TF).

We consider the ring $A = \mathbf { Z } [ X , Y ]$. Let $U$ be the set of elements of $A$ of the form $X Y ^ { k }$ with $k \in \mathbf { N }$, we set $B = \mathcal { A } ( U )$. Let $S$ be a finite subset of $B$. a) Show that there exists $m \in \mathbf { N } ^ { * }$ such that $\mathcal { A } ( S ) \subset \mathcal { A } \left( \left\{ X , X Y , \ldots , X Y ^ { m } \right\} \right)$. b) Show that there exists an integer $N > 0$ such that every element of $\mathcal { A } ( S )$ is a sum of monomials of the form $\alpha X ^ { i } Y ^ { j }$ with $\alpha \in \mathbf { Z }$ and $j \leq i N$. c) Deduce that the ring $B$ does not have property (TF).

Let $E$ be a finite subset of $M _ { n } ( A )$. Show that there exists a subring $B$ of $A$ such that: $B$ has property (TF) and for every matrix $M \in E$, all coefficients of $M$ belong to $B$.

Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field. a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$. b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral. c) Deduce that the result of a) still holds for every commutative ring $A$. d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have $$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$