When is irreducibility of a polynomial over a field equivalent to not having any roots in it?
Apart from of course, the simple cases when a polynomial $f \in K[x]$ is of degree less than or equal to three. One direction is clear: If a polynomial is irreducible in $K$, it can have no roots in it. But the converse is much more bizarre. So I pose:
- What conditions must be put on $f$ so that this happens?
- How does information about $K$ alter this?
I do not even know where to start. Links to any research done in this area is also appreciated :) Thanks, guys!