Let's say we have a prime number $p=5,$ and we are interested in how the residue field $\mathcal{O}_k/P$ looks like, where $P$ lies over 5 in quadratic extension $k.$
I suppose this has to depend on whether 5 ramifies or not in the extension, but I'm not quite sure how to proceed with that.
Also, I saw somewhere that the residue field can be $\mathbb{F}_5$ or $\mathbb{F}_{25}$ but I'm also not sure how to get to that.
Edit: what about cubic extensions?