Given a set $\{(p_1, e_1),(p_2, e_2)\dots(p_n, e_n)\}$, does there exist a Galois extension $R$ of the natural numbers such that $p_i$ splits into $\frac{k}{e_i}$ ideals for all $i$ and some $k$?
In this instance, I'm happy to treat a prime that is ramified with index $m$ as a prime that splits into $m$ ideals - I'm concerned with the size of the ideals rather than the number of ideals generated.
Sorry if the specific terms for the object I want are wrong; I might actually be looking for the integral closure of a Galois extension of the rational numbers. Alternately, I might be looking for a ring of integers in a number field with class number 1 (and those definitions might be equivalent; I'm not 100% sure). In any case, $R$ should be a Dedekind domain and the size of $\frac{R}{\langle p_i\rangle}$ should be $p_i^{\frac{k}{e_i}}$.
