Let $G$ be a finite group.
(1) Let $K$ be a field. Show that $G$ has a finite-dimensional fait hful $K$-linear representation.
(2) Show that $G$ has a faithful one-dimensional complex repr esentation if and only if $G$ is cyclic.
(3) Assume moreover that $G$ is commutative. Let $n \geq 1$ be an integer. Show that $G$ has a faithful $n$-dimensional complex re presentation if and only if $G$ can be generated by $n$ elements.
(4) Classify all finite groups having a faithful 2-dimensional re al representation.

Let $n \geq 1$ be an integer. Let $A$ be a discrete valuation ring wit h $K$ its field of fractions and $\pi \in A$ a uniformizer. For $\lambda=\left(\lambda_1, \cdots, \lambda_n\right) \in \mathbb{Z}^n$ write
$$
D_\lambda=\operatorname{diag}\left(\pi^{\lambda_1}, \cdots, \pi^{\lambda_n}\right)=\left(\begin{array}{cccc}
\pi^{\lambda_1} & 0 & \cdots & 0 \\
0 & \pi^{\lambda_2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \pi^{\lambda_n}
\end{array}\right) \in \mathrm{GL}_n(K) .
$$

Show that, for $\lambda, \mu \in \mathbb{Z}^n$, the following intersection inside $\mathrm{GL}_n(K)$
$$
\mathrm{GL}_n(A) \cdot D_\mu \cdot \mathrm{GL}_n(A) \bigcap U(K) \cdot D_\lambda
$$
is non-empty if and only if $\lambda_{\text {dom }} \leq \mu_{\text {dom }}$. Here
- $\mathrm{GL}_n(K)$ (resp. $\mathrm{GL}_n(A)$ ) is the group of invertible $n \times n \mathrm{sq}$ uare matrices with coefficients in $K$ (resp.in $A$ ), and $U(K) \subset \mathrm{GL}_n(K)$ is the standard unipotent subgroup, that $\mathrm{i}$ $\mathrm{s}$, the subgroup of upper triangular matrices with coefficients 1 on the diagonal.
- for $\alpha=\left(a_1, \cdots, a_n\right)$ and $\beta=\left(b_1, \cdots, b_n\right)$ two elements $\mathrm{i}$ $\mathrm{n} \mathbb{Z}^n$, we write $\alpha \leq \beta$ if
$$
\sum_{i=1}^k a_i \leq \sum_{i=1}^k b_i, \quad \text { for any } 1 \leq k \leq n
$$
and if $\sum_{i=1}^n a_i=\sum_{i=1}^n b_i$. Write also $\alpha_{\text {dom }}:=\left(a_1^{\prime}, \cdots, a_n^{\prime}\right)$ with $a_1^{\prime}, \cdots, a_n^{\prime}$ an arrangement of $a_1, \cdots, a_n$ such that
$$
a_1^{\prime} \geq a_2^{\prime} \geq \cdots \geq a_n^{\prime}
$$

Let $k$ be an imperfect field of characteristic $p>0$. Let $a \in k \backslash k^p$.
(1) Show that the polynomial $X^p-a \in k[X]$ is irreducible.
(2) Let $A=k[X] /\left(X^{p^2}-a X^p\right)$. Compute $A_{\text {red }}$, the quotien $\mathrm{t}$ of $A$ by its nilpotent radical.

Let $k$ be a field of character $p>0$ and consider $k(t) \subset k((t))$.
(1) Show that the field extension $k((t)) / k(t)$ is transcendent al, i.e., $k((t))$ contains at least one transcendental element ov er $k(t)$. (Hint: don't spend too much time on this question.)
(2) Let $\alpha \in k[[t]]$ which is transcendental over $k(t)$, and write $\beta=\alpha^p \in k[[t]]$. Let $K=k(t, \beta)$ and $L=k(t, \alpha)$ : both are subfields of $k((t))$. Let $A=k[[t]] \cap k(t, \beta)$. Determine the i ntegral closure $B$ of $A$ in $L$, and show that $A$ and $B$ are two discrete valuation rings.
(3) Is $B$ finitely generated over $A$ as a module? Justify your as sertion.

Let $p$ be a prime number. Consider $\mathbb{Z}_p$ (resp. $\mathbb{Q}_p$ ) the ring of $p$ adic integers (resp. field of $p$-adic numbers). Clearly $\mathbb{Z} \subset \mathbb{Z}_p$.
(1) Show that the set $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, and deduce that a ma $\mathrm{p} f: \mathbb{Z} \rightarrow \mathbb{Q}_p$ can be extended to a continuous function on $\mathbb{Z}_p$ if and only if $f$ is uniformly continuous, i.e., for any $\varepsilon>0$. there exists some integer $N>0$ so that $|f(n)-f(m)| < \varepsilon$ for any integers $n, m \in \mathbb{Z}$ with $m \equiv n\left(\bmod p^N\right)$.
(2) Let $a \in \mathbb{Q}_p \backslash\{0\}$. Under what condition on $a$, the map
$$
f: \mathbb{Z} \rightarrow \mathbb{Q}_p, \quad n \mapsto a^n
$$
can be extended to a continuous function over $\mathbb{Z}_p$ ? Justify yo ur assertion.
(3) Assume that the condition in (2) is fulfilled. Can we even $\mathrm{e}$ xtend the function $f$ in (2) to a continuous map
$$
a^x: \mathbb{Q}_p \rightarrow \mathbb{Q}_p,
$$
so that $a^{x+y}=a^x a^y$ for any $x, y \in \mathbb{Q}_p$ ? Justify your assertio n.

Let $n \geq 1$ be an integer and write $\Phi_n(X)$ the $n$-th cyclotomi c polynomial, that is, the minimal polynomial of a primitive $n$ th root of unity in $\mathbb{C}$ over $\mathbb{Q}$. Write also $\varphi(n)=\operatorname{deg}\left(\Phi_n(X)\right)$
(1) Let $q$ be a power of a prime number such that $(q, n)=1$. Show that $\Phi_n$, viewed as an element in $\mathbb{F}_q[X]$, can be decom posed as a product of $\varphi(n) / d$ irreducible polynomials of deg ree $d$, with $d$ the order of $q$ in the multiplicative group $(\mathbb{Z} / n \mathbb{Z})^{\times}$.
(2) From now on, assume $n=2^{r+1}$ for some integer $r \geq 1$. L et $\zeta=\zeta_n$ be a primitive $n$-th root of unity and $K=\mathbb{Q}[\zeta]$. Let $p$ be a prime with $p \equiv-3(\bmod 8)$.
(a) For $x, y \in K=\mathbb{Q}[\zeta]$, define
$$
(x, y):=\sum_\tau \tau(x) \cdot \overline{\tau(y)}
$$
where $\tau$ runs through all the embeddings $K \hookrightarrow \mathbb{C}$ of $K$ into the field $\mathbb{C}$ of complex numbers. Write $K_{\mathbb{R}}=K \otimes_{\mathbb{Q}} \mathbb{R}$, and we use the same notation to denote the (à priori $\mathbb{C}$-valued) bi linear form on $K_{\mathbb{R}}$ obtained by extension of scalars. Show tha $\mathrm{t}(\cdot, \cdot)$ gives an inner product on $K_{\mathbb{R}}$ and for $0 \leq i, j < 2^r$,
$$
\left(\zeta^i, \zeta^j\right)= \begin{cases}2^r, & \text { if } i=j \\ 0, & \text { otherwise }\end{cases}
$$

In particular, we obtain an Euclidean space $K_{\mathbb{R}}$ and $\left(\zeta^i / \sqrt{2^r}\right)_{0 \leq i < 2^r}$ is an orthonormal basis.
(b) Decompose $p \mathcal{O}_K$ into a product of prime ideals.
(c) Let $\mathfrak{p} \subset \mathcal{O}_K$ be a prime ideal of $\mathcal{O}_K$ containing $p$. Show th at for every $\alpha \in \mathfrak{p},|\alpha|^2 \in 2^r p \mathbb{Z}$, and compute the length of $\mathrm{t}$ he shortest non-zero vector in the prime ideal $\mathfrak{p} \subset K_{\mathbb{R}}$.