The following problem is from Abstract Algebra, Herstein. Problem 10 in Chapter 5.6. I'm trying to self-study abstract algebra using this book and am unsure how to think about extension fields. I assume there are many answers to this problem. Any answers would give me an idea of what type of examples to think about when thinking about finite extensions.
Construct an extension field $K_n$ of $\mathbb{Q}$ such that $[K_n : \mathbb{Q}] = n$, for any $n \geq 1$.
Pick an irreducible polynomial $f$ of degree $n$. For example $f(x)=x^n+p$, for some prime $p$ and extend $\mathbb{Q}$ by one of the roots $r$ of $f$.
Then $K_n=\mathbb{Q}(r)$ will have $1,r,r^2,...,r^{n-1}$ as a basis over $\mathbb{Q}$.
In fact, they are linearly independent, because otherwise a linear dependence between them will result in a polynomial over $\mathbb{Q}$ of which $r$ is a root, but of degree strictly smaller than $n$. Such a polynomial would have a common factor with $f$ over $\mathbb{Q}$, contradicting that $f$ is irreducible.
They generate $K_n$ because any polynomial $g(r)$ can be expressed as $f(r)q(r)+s(r)$ with degree of $s<n$.