Minimum degree for an irreducible polynomial

Question

Is it true that for every non-closed field $\mathbb F$ exists a polynomial of degree 2 that is irreducible over $\mathbb F$? This holds for $\mathbb Q, \mathbb R$ but I can't understand if it is always true.

Answer 1

No. See for instance the field of straightedge-and-compass (complex) constructible numbers.

Another example is like this: consider $\overline {\Bbb F}_p$, inside which you have exaclty one copy of each $\Bbb F_{p^n}$. Then, $\bigcup_{m\in\Bbb N}\Bbb F_{p^{2^{m}}}$ is a subfield and it's as you mention.