Consider the following number field $K = \frac{\mathbb{Q}[x]}{(x^{(p^2-1)/2}-p)}$ where $p$ is a prime. What is the factorization of the prime ideal $p$ in the ring of integers of $K$?



Ramification of a prime in a number field
If you are interested only in the ramification above a single prime number $p$, it is best to forget the field $\mathbf{Q}$ and work instead over the field $\mathbf{Q}\_p$. For every $n>0$, the polynomial $f=x^n-p$ is an Eisenstein polynomial over $\mathbf{Q}\_p$, so it is irreducible, and adjoining a root of $f$ gives a totally ramified extension of degree $n$.


