Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function such that $f(z+1) = f(z+\imath) = f(z)$ for every $z \in \mathbb{C}$. Choose the correct statement(s) from below:
(A) $f$ is constant;
(B) $f(z) = 0$ for every $z \in \mathbb{C}$;
(C) There exist complex numbers $a, b$ such that for every $x, y \in \mathbb{R}$, $f(x + \imath y) = a\sin(x) + \imath b\cos(y)$;
(D) $f$ is not necessarily constant but $|f(z)|$ is constant.

Consider the $\mathbb{Q}$-vector-space $$\{ f : \mathbb{R} \longrightarrow \mathbb{R} \mid f \text{ is continuous and } \mathrm{Image}(f) \subseteq \mathbb{Q} \}$$ What is its dimension?

Let $p$ be a prime number and $F$ a field of $p^{23}$ elements. Let $\phi : F \longrightarrow F$ be the field automorphism of $F$ sending $a$ to $a^p$. Let $K := \{ a \in F \mid \phi(a) = a \}$. What is the value of $[K : \mathbb{F}_p]$?

Let $A$ be a non-trivial subgroup of $\mathbb { R }$ generated by finitely many elements. Let $r$ be a real number such that $x \longrightarrow r x$ is an automorphism of $A$. Show that $r$ and $r ^ { - 1 }$ are zeros of monic polynomials with integer coefficients.

Throughout the rest of the problem, we fix an odd integer $\ell \geq 3$ and $q$ a primitive $\ell$-th root of unity. Show that $q^2$ is a primitive $\ell$-th root of unity.

Let $d \in \mathbb { N } ^ { * }$. Construct a map $$\left\lvert \, \begin{aligned} r : \quad \mathbb { R } _ { d } [ X ] & \rightarrow \mathbb { R } \\ P & \mapsto r ( P ) \end{aligned} \right.$$ polynomial in the coefficients of $P$, such that, if $r ( P )$ is nonzero, then the roots of $P$ in $\mathbb { C }$ are simple.
Hint: You may use the previous question.

Let $d \in \mathbb { N } ^ { * }$ and $f$ a polynomial function on $\mathbb { R } ^ { d }$. Suppose that the function $f$ is nonzero. Show that $f ^ { - 1 } ( \mathbb { R } \backslash \{ 0 \} )$ is dense in $\mathbb { R } ^ { d }$.
Hint: You may use the fact that a nonzero polynomial in one variable has only finitely many roots.

Show that $I ( \alpha )$ is an ideal of $\mathbb { Q } [ X ]$, different from $\{ 0 \}$, where $\alpha$ is a fixed algebraic number and $I ( \alpha ) = \{ P \in \mathbb { Q } [ X ] \mid P ( \alpha ) = 0 \}$.

Show that $\alpha$ has degree 1 if and only if $\alpha \in \mathbb { Q }$, where the degree of $\alpha$ is the degree of its minimal polynomial $\Pi_{\alpha}$.

(a) Show that $\Pi _ { \alpha }$ is irreducible in $\mathbb { Q } [ X ]$.
(b) Let $P \in \mathbb { Q } [ X ]$ be a monic polynomial, irreducible in $\mathbb { Q } [ X ]$. Show that if $z$ is a complex root of $P$, then $P$ is the minimal polynomial of $z$.

(a) Show that if $\alpha \in \mathbb { Q }$ is an algebraic integer, then $\alpha \in \mathbb { Z }$.
(b) Show that if $\alpha \in \mathbb { C }$ is an algebraic integer then $\Pi _ { \alpha } \in \mathbb { Z } [ X ]$.
Hint: use the theorem admitted in the introduction (the set of algebraic integers is a subring of $\mathbb{C}$) as well as question 5a.

(a) Let $\alpha \in \mathbb { C }$ be an algebraic integer of degree 2 and of modulus 1. Show that $\alpha$ is a root of unity.
(b) Show that $\frac { 3 + 4 i } { 5 }$ is an algebraic number of degree 2 and of modulus 1 but is not a root of unity.

Show that for all $n \geq 1$ we have $$X ^ { n } - 1 = \prod _ { d \mid n } \Phi _ { d }$$ the product being taken over the set of positive integers $d$ dividing $n$, where $\Phi_n = \prod_{z \in \mathbb{P}_n}(X - z)$ and $\mathbb{P}_n$ is the set of primitive $n$-th roots of unity.

(a) Show that if $p$ is a prime number and $k \geq 1$ is an integer, then $$\Phi _ { p ^ { k } } = X ^ { ( p - 1 ) p ^ { k - 1 } } + X ^ { ( p - 2 ) p ^ { k - 1 } } + \cdots + X ^ { p ^ { k - 1 } } + 1$$ (b) Calculate $\Phi _ { n }$ for $n = 1,2,3,4,5,6$.

We fix an integer $n \geq 2$.
(a) Calculate $\Phi _ { n } ( 0 )$.
(b) Calculate $\Phi _ { n } ( 1 )$ as a function of the prime factorization of $n$. Hint: reason by induction on $n$, using question 7.

Show that $\Phi _ { n } \in \mathbb { Z } [ X ]$.

Let $P \in \mathbb { Z } [ X ]$ be a monic polynomial of degree $n \geq 1$, irreducible in $\mathbb { Q } [ X ]$ and all of whose complex roots have modulus 1. Let $z _ { 1 } , \ldots , z _ { n }$ be the complex roots of $P$ counted with their multiplicities, so that $P = \prod _ { i = 1 } ^ { n } \left( X - z _ { i } \right)$. For every integer $k \geq 0$ we denote $a _ { k } = z _ { 1 } ^ { k } + z _ { 2 } ^ { k } + \cdots + z _ { n } ^ { k }$.
(a) Show that the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$ converges for all $z \in \mathbb { C }$ such that $| z | < 1$.
(b) Let $z \in \mathbb { C }$ be non-zero such that $| z | < 1$ and let $f ( z )$ be the sum of the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$. Show that $$z f ( z ) P \left( \frac { 1 } { z } \right) = P ^ { \prime } \left( \frac { 1 } { z } \right)$$ (c) Deduce that $a _ { k } \in \mathbb { Z }$ for all $k \geq 0$.

Let $P \in \mathbb { Z } [ X ]$ be a monic polynomial of degree $n \geq 1$, irreducible in $\mathbb { Q } [ X ]$ and all of whose complex roots have modulus 1. Let $z _ { 1 } , \ldots , z _ { n }$ be the complex roots of $P$ counted with their multiplicities. For every integer $k \geq 0$ we denote $a _ { k } = z _ { 1 } ^ { k } + z _ { 2 } ^ { k } + \cdots + z _ { n } ^ { k }$.
(a) Show that there exist two integers $0 \leq k < l$ such that $a _ { k + i } = a _ { l + i }$ for all $i \in \{ 0,1 , \ldots , n \}$. We fix two such integers $k , l$ in questions 12b and 12c.
(b) Show that $\sum _ { i = 1 } ^ { n } F \left( z _ { i } \right) \left( z _ { i } ^ { l } - z _ { i } ^ { k } \right) = 0$ for every polynomial $F \in \mathbb { C } [ X ]$ of degree at most $n$.
(c) Show that $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are pairwise distinct. Deduce that $z _ { i } ^ { l - k } = 1$ for all $i \in \{ 1,2 , \ldots , n \}$ and conclude.

Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Let $F , G \in \mathbb { Z } [ X ]$. Show that there exists $H \in \mathbb { Z } [ X ]$ such that $$( F + G ) ^ { p } = F ^ { p } + G ^ { p } + p H$$ (b) Show that $\Pi _ { z } \in \mathbb { Z } [ X ]$ and deduce the existence of a polynomial $F \in \mathbb { Z } [ X ]$ such that $$\Pi _ { z } \left( X ^ { p } \right) = \Pi _ { z } ( X ) ^ { p } + p F ( X )$$ (c) Show that $\frac { \Pi _ { z } \left( z ^ { p } \right) } { p }$ is an algebraic integer.

Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Express as a function of $n$ the number $\prod _ { 1 \leq i < j \leq n } \left( z _ { i } - z _ { j } \right) ^ { 2 }$, where $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are the roots of the polynomial $P = X ^ { n } - 1$. Hint: One may consider the numbers $P ^ { \prime } \left( z _ { i } \right)$.
(b) Show that $\Pi _ { z } \left( z ^ { p } \right) = 0$. Hint: show that if $\Pi _ { z } \left( z ^ { p } \right) \neq 0$, then there exists an algebraic integer $u$ such that $n ^ { n } = u \cdot \Pi _ { z } \left( z ^ { p } \right)$.
(c) Conclude that $\Phi _ { n } = \Pi _ { z }$.

We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Let $\alpha$ be an element of $\mathcal { S }$ and let $\gamma \in C ( \alpha )$ have modulus 1.
(a) Show that the minimal polynomial of $\alpha$ is reciprocal and that $\frac { 1 } { \alpha }$ is a conjugate of $\alpha$.
(b) Show that $\gamma$ is not a root of unity.
(c) Show that all conjugates of $\alpha$ other than $\frac { 1 } { \alpha }$ have modulus 1.

We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Show that the degree of every element of $\mathcal { S }$ is an even integer, greater than or equal to 4.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Verify that $P _ { n }$ has no root in $\mathbb { Q }$ and that $P _ { n }$ has at least one real root strictly greater than 1. We fix such a root $\alpha _ { n }$ in the sequel.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $x$ be a complex number. Show that $x$ is totally real if and only if $x ^ { 2 }$ is totally positive.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $x$ be a complex number. Show that $x$ is totally real if and only if $x ^ { 2 }$ is totally positive.