I am struggling with the following problem:
Let $L/K$ be a field extension and $x \in L$ be transcendental. Show that the field extension $K(x^n) \subset K(x)$ is algebraic of degree $n$. Determine the minimal polynomial of $x$ over $K(x^n)$
So, it is pretty obvious that the degree cannot be larger than $n$ since we have the polynomial $$p(t)=t^n-x^n$$ which has $x$ as a root. And now it would either be sufficient to show that $p$ is irreducible or that the degree is greater or equal than $n$. I have tried both but the crucial point is that I do not really know how the elements in $K(x^n)$ look like since it is not an algebraic extension where we know the basis.