I am trying to solve a problem from Dummit-Foote's Section 14.9 (problem 15):
Let $K_0= \mathbb Q$ and for $n > 0$ define the field $K_{n+1}$ as the extension of $K_n$ obtained by adjoining to $K_n$ all roots of all cubic polynomials over $K_n$. Let $K$ be the union of the subfields $K_n$. Prove that there are nontrivial algebraic extensions of $K$.
I have tried a lot but can't figure out how to show this. I'm guessing that attaching roots of some irreducible quartic polynomial over $\mathbb Q$ might give a contradiction, but I am stuck. Can someone please provide some hints?