True or false: there exists a field $K$ and a positive integer $n$ such that every polynomial over $K$ of degree $n$ is reducible (in $K[z]$) but there exists an irreducible polynomial of degree $n+1$.
Motivation: two extreme cases are well known. If $K={\mathbb Q}$ the statement is false as there are irreducible polynomials of every degree. I am not sure but I believe this is also true for finite fields. If $K={\mathbb R}$ then there are irreducible polynomials of degree $1$ and $2$ only.
$\bigcup_{m\ge 1}\Bbb{F}_{p^{2^m}}$ doesn't have any quadratic extension.
The quadratic closure of $\Bbb{Q}$ works too: that is $E=\bigcup_{m\ge 0} E_m$ where $E_0=\Bbb{Q},E_{m+1}=E_m(\{ a^{1/2},a\in E_m\})$. Any element $b\in E$ is contained in a finite tower of quadratic extensions so $[\Bbb{Q}(b):\Bbb{Q}]$ divides a power of $2$, therefore $x^3-2$ is irreducible over $E$.