Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient. [b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $ [b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $

Let $n$ and $q$ be two natural numbers, $n\ge 2$, $q\ge 2$ and $q\not\equiv 1 (\text{mod}\ 4)$ and let $K$ be a finite field which has exactly $q$ elements. Show that for any element $a$ from $K$, there exist $x$ and $y$ in $K$ such that $a = x^{2^n} + y^{2^n}$. (Every finite field is commutative).

Let $q > 1$ be a power of $2$. Let $f: \mathbb{F}_{q^2} \to \mathbb{F}_{q^2}$ be an affine map over $\mathbb{F}_2$. Prove that the equation \[ f(x) = x^{q+1} \] has at most $2q - 1$ solutions.

Let $S$ be the set of $2\times2$-matrices over $\mathbb{F}_{p}$ with trace $1$ and determinant $0$. Determine $|S|$.

Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by \[\tau(f)(x)=f(x+1).\] Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

Prove that $p(x)=1+x+x^2 +\ldots+x^n$ is reducible over $\mathbb{F}_{2}$ in case $n+1$ is composite. If $n+1$ is prime, is $p(x)$ irreducible over $\mathbb{F}_{2}$ ?

Consider a finite field $ K. $ [b]a)[/b] Prove that there is an element $ k $ in $ K $ having the property that the polynom $ X^3+k $ is irreducible in $ K[X], $ if $ \text{ord} (K)\equiv 1\pmod {12}. $ [b]b)[/b] Is [b]a)[/b] still true if, intead, $ \text{ord} (K) \equiv -1\pmod{12} ? $ [i]Dorel Miheț[/i]

Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.

Let K be the field formed by the addition of a root of the polynomial $x^4 - 2x^2 - 1$ to the rational field. Prove that there are no exceptional units in the ring of integers of K. (A unit $\varepsilon$ is called exceptional if $1-\varepsilon$ is also a unit.)

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$

Let $\mathbb{L}$ be a finite field with $q$ elements. Prove that: a) If $q \equiv 3 \pmod 4$ and $n \ge 2$ is a positive integer divisible by $q-1,$ then $x^n=(x^2+1)^n$ for all $x \in \mathbb{L}^{\times}.$ b) If there exists a positive integer $n \ge 2$ such that $x^n=(x^2+1)^n$ for all $x \in \mathbb{L}^{\times},$ then $q \equiv 3 \pmod 4$ and $q-1$ divides $n.$