Here's an example of historical importance for $n=2$:
Let $F$ be the subfield of $\mathbb C$ consisting of all numbers constructible with ruler and compass if $0$, $1$ and $i$ are given. ($F$ is a field because adding, negating, multiplying, taking reciprocals can be done with ruler and compass).
Since it is possible to take square roots and solve quadratics with ruler and compass, we see that all quadratic polynomials are reducible.
On the other hand, the old classic problem of doubling the cube (i.e. finding a root of $x^3-2=0$) is not solvable with ruler and compass, hence $F\ne \overline F$.