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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!

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    an obvious thing: if there are no irreducible polynomials of degree $2$ (i.e. no extensions of $K$ of degree $2$) then you have your property for polynomials of degree $4$. Etc.2012-09-07
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    In fact, if there are no degree 2 irreducibles, property holds in degree 5 as well.2012-09-07
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    What are examples of non algebraically closed fields with no degree $2$ extensions?2012-09-07
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    @Georges, I don't know, but, reasoning naively, if you start with the rationals, and throw in square roots of all rationals, and then square roots of everything in *that* field, and iterate to infinity, you ought to reach a field with no quadratic extensions but not algebraically closed because not containing, for example, the cube root of 2.2012-09-07
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    Dear @Gerry: yes, that's an excellent idea, thank you.2012-09-07

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