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.

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.

Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statements are equivalent.
(A) For every nontrivial subgroup $K$ of $G$, the subgroup $K \cap H$ is also nontrivial.
(B) $H$ contains every nontrivial minimal subgroup of $G$ and every element of the quotient group $G / H$ has finite order.

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 $X \subseteq \mathbb { R } ^ { n }$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma ^ { \prime } ( 0 )$, where $\gamma : ( - \epsilon , \epsilon ) \longrightarrow X$ is a differentiable function with $\gamma ( 0 ) = p$. ($\epsilon \in \mathbb { R } , \epsilon > 0$.) The tangent space of $X$ at $p$ is the $\mathbb { R }$-vector space of all the tangent vectors at $p$. Think of $\mathrm { GL } _ { n } ( \mathbb { C } )$ as a subspace of $\mathbb { R } ^ { 2 n ^ { 2 } }$, with the euclidean topology.
Let $G : = \left\{ A \in \mathrm { GL } _ { 2 } ( \mathbb { C } ) \mid A ^ { * } A = A A ^ { * } = I _ { 2 } , \operatorname { det } A = 1 \right\}$.
(A) (2 marks) Show that every tangent vector of $\mathrm { GL } _ { n } ( \mathbb { C } )$ at $I _ { n }$ is of the form $\gamma _ { A } ^ { \prime } ( 0 )$ where $A$ is a $n \times n$ complex matrix and $\gamma _ { A } : \mathbb { R } \longrightarrow \mathrm { GL } _ { n } ( \mathbb { C } )$ is the function $t \mapsto e ^ { t A }$.
(B) (3 marks) Show that the tangent space of $G$ at $I _ { 2 }$ is $V : = \left\{ \left. \left[ \begin{array} { c c } i a & z \\ - \bar { z } & - i a \end{array} \right] \right\rvert \, a \in \mathbb { R } , z \in \mathbb { C } \right\}$.
(C) (5 marks) Consider the homeomorphism $\Phi : G \longrightarrow \mathbb { S } ^ { 3 }$ (where $\mathbb { S } ^ { 3 }$ denotes the unit sphere in $\mathbb { R } ^ { 4 }$) given by $$\left[ \begin{array} { c c } \alpha & \beta \\ \bar { \beta } & \bar { \alpha } \end{array} \right] \mapsto ( \Re ( \alpha ) , \Im ( \alpha ) , \Re ( \beta ) , \Im ( \beta ) )$$ Define a 'multiplication' on $V$ by $[ A , B ] = \frac { A B - B A } { 2 }$. Determine the multiplication on the tangent space at $\Phi \left( I _ { 2 } \right)$ induced by the derivative $D \Phi$. (Hint: The map $( A , B ) \longrightarrow [ A , B ]$ is $\mathbb { R }$-bilinear.)

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 $\operatorname { deg } \left( Q _ { i } \right) = d$.

Show that the set of non-zero similarities is a subgroup of $GL(E)$ under composition of applications.

Let $h \in \mathscr{L}(E)$ be an endomorphism of $E$. Show that the following properties are equivalent: i) $h$ is an element of $\operatorname{Sim}(E)$; ii) $h^{*}h$ is collinear to $\operatorname{Id}_{E}$; iii) the matrix of $h$ in an orthonormal basis of $E$ is collinear to an orthogonal matrix.

Let $f$ be an antisymmetric endomorphism of $E$. Show that: $\forall x \in E, \langle x, f(x) \rangle = 0$.

Let $f$ be an antisymmetric endomorphism of $E$. Show that, if $S$ is a vector subspace of $E$ stable under $f$, then $S^{\perp}$ is stable under $f$. Show that the endomorphisms induced by $f$ on $S$ and on $S^{\perp}$ are antisymmetric.

Let $f$ be an antisymmetric endomorphism of $E$. Let $g$ be an antisymmetric endomorphism of $E$, such that $fg = -gf$. Show that: $\forall x \in E, \langle f(x), g(x) \rangle = 0$.

Let $f$ be an antisymmetric endomorphism of $E$. What is $f^{2} = f \circ f$ if $f$ is an orthogonal automorphism and antisymmetric of $E$?

Show that $d_{n} \geqslant 1$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$. We fix $x \in E \backslash \{0\}$. By considering $\Phi : f \mapsto f(x)$, linear application from $V$ to $E$, show that $\operatorname{dim}(V) \leqslant n$. Thus $1 \leqslant d_{n} \leqslant n$.

In this question only, we assume $n = 2$. Explicitly give a vector space of dimension 2, formed of similarity matrices. Deduce from this, carefully, that $d_{2} = 2$.

In this question only, we assume $n$ is odd. If $f, g$ belong to $GL(E)$, show that there exists $\lambda \in \mathbb{R}$ such that $f + \lambda g$ is non-invertible. One may reason by considering the characteristic polynomial of $fg^{-1}$. Deduce that $d_{n} = 1$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of dimension $d \geqslant 1$. Show that there exists a vector subspace $W$ of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of the same dimension $d$, and containing $\operatorname{Id}_{E}$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that for all $i \in \{1, 2, \ldots, d-1\}$, $f_{i}^{*} + f_{i}$ is collinear to $\operatorname{Id}_{E}$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that there exists a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ such that for all $i \in \{1, 2, \ldots, d-1\}$, $g_{i}$ is antisymmetric (one will seek $g_{i}$ as a combination of $f_{i}$ and $\operatorname{Id}_{E}$).

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. We fix a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ with for all $i$, $g_{i}$ antisymmetric. a) Show that for all $i \neq j$, $g_{i}g_{j} + g_{j}g_{i}$ is collinear to $\operatorname{Id}_{E}$. b) Show that we define a scalar product on $\mathscr{L}(E)$ by setting, for all $f, g$ of $\mathscr{L}(E)$: $(f \mid g) = \operatorname{tr}(f^{*}g)$. We consider, in the rest of this question, a basis $(h_{1}, \ldots, h_{d-1})$ of $\operatorname{Vect}(g_{1}, \ldots, g_{d-1})$ orthogonal for this scalar product. c) Show that the $h_{i}$ are antisymmetric and satisfy: $\forall i \neq j, h_{i}h_{j} + h_{j}h_{i} = 0$. What should be done so that the $h_{i}$ are also orthogonal automorphisms?

Conversely, let $(h_{1}, \ldots, h_{d-1})$ be a family of $\mathscr{L}(E)$ such that the $h_{i}$ are antisymmetric orthogonal automorphisms satisfying for all $i \neq j$, $h_{i}h_{j} + h_{j}h_{i} = 0$. Show that $\operatorname{Vect}\left(\operatorname{Id}_{E}, h_{1}, \ldots, h_{d-1}\right)$ is a vector subspace of $\mathscr{L}(E)$, of dimension $d$, included in $\operatorname{Sim}(E)$.

Let $p$ be an odd integer such that $\operatorname{dim}(E) = n = 2p$. We assume that there exist $d \geqslant 3$ and a family $(f_{1}, f_{2}, \ldots, f_{d-1})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let $x \in E$ of norm 1. a) Show that $(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is an orthonormal family, and that $S = \operatorname{Vect}(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is stable under $f_{1}$ and $f_{2}$. b) Deduce that $d_{n-4} \geqslant 3$.

In this question, $n = 2p$, with $p$ an odd integer. Show that $d_{n} = 2$.

In this section, the dimension of $E$ is 4. We assume that there exists a vector subspace of $\mathscr{L}(E)$ of dimension 4 included in $\operatorname{Sim}(E)$. We then consider, in accordance with I.D.4, a family $(f_{1}, f_{2}, f_{3})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let a fixed vector $x \in E$ of norm 1. a) Justify that the family $B = (x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is a basis of $E$ then show that there exist real numbers $\alpha, \beta, \gamma, \delta$ such that: $$f_{3}(x) = \alpha x + \beta f_{1}(x) + \gamma f_{2}(x) + \delta f_{1}f_{2}(x)$$ Show that $\alpha = \beta = \gamma = 0$ and that $\delta \in \{-1, 1\}$. b) Show that $f_{3} = \delta f_{1}f_{2}$. If necessary, by replacing $f_{3}$ with its opposite, we assume in what follows that $f_{3} = f_{1}f_{2}$. c) If $x_{0}, x_{1}, x_{2}, x_{3}$ are real numbers, give the matrix $M(x_{0}, x_{1}, x_{2}, x_{3})$ in $B$ of the endomorphism $x_{0}\operatorname{Id}_{E} + x_{1}f_{1} + x_{2}f_{2} + x_{3}f_{3}$.

In this section, the dimension of $E$ is 4. Verify that for all $(x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4}$, $M(x_{0}, x_{1}, x_{2}, x_{3})$ is a similarity matrix. What can we conclude?