Found problems: 7
Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \ldots, n_p$. Prove that the polynomial
\[\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}\]
is irreducible in $\mathbb{Q}[X]$.
[i]Beniamin Bogosel[/i]
Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.
For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that
$$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.
If at least one of the integers $ a,b $ is not divisible by $ 3, $ then the polynom $ X^2-abX+a^2+b^2 $ is irreducible over the integers.
[i]Ion Cucurezeanu[/i]
Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that:
(a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$;
(b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.
Prove that for each $n\in\mathbb N$ the polynomial $(x^2+x)^{2^n}+1$ is irreducible over the polynomials with integer coefficients.
Let $f(x) = x^4 + 2x^3 - x - 1$.
(a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.
(b) Find the exact values of the 4 roots of $f(x)$.