Deleted the old question, because tho whole question kind of changed. I am facing following problem:
Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ with coefficients in $K$ does have root in $L$.
I know now the proof using some knowledge about splitting fields of irreducible polynomials. However, I am trying now to do somewhat more elementary and quite nonelegant proof and I would be very glad if someone would have look at this. I don't know whether a) what I have so far is correct b) even if it is, whether it leads anywhere...
edit: thanks to both of you guys, I finally got the exact number of irreducible polynomials in $K[x]$ right, which should be $(q^3 - q)/3$. Still working on how to use this number to finish the original problem.
note: the extension is finite, hence algebraic, so every element of $F$ is root of some polynomial in $K[x]$. So in particular, I have $q^3 - q$ "new" roots.Above mentioned $(q^3 - q)/3$ irreducible monic polynomials can have at most $q^3 - q$ roots in any extension. This "new" roots are exactly the roots of those polynomials, if I am not mistaken, but I don't know how to prove that.
Each element of $L$ that isn't in $K$ has a minimal polynomial of degree $3$. At most three of them can share the same minimal polynomial.
You may wish to count more accurately: e.g. you're only counting $x^3$ as one sixth of a polynomial.