I've been making a light study of the relationships between Diophantine Equations and their relations to class Groups of Algebraic Number Fields. For the most part (aside from degree two) there is no direct relationship however an interesting aside question comes up (assuming my math isn't just plain inaccurate)

Consider the set of all ideals in an algebrainc number field $K/\mathbb{Q}$

For each prime $p \in \mathbb{Z}$ let us choose exactly one prime above $\mathfrak{P} | (p) $ in $O_K$

Consider then $I'_K$ to be the subset of the fractional Ideal Group $I_K$ generated entirely by this subset of all the ideals in $K$

So if this is not wrong thinking my question is what would the group $I'_K/P'_K$ be ?

I've been making a light study of the relationships between Diophantine equations and their relations to class groups of algebraic number fields. For the most part (aside from degree two) there is no direct relationship however an interesting aside question comes up (assuming my math isn't just plain inaccurate)

Consider the set of all ideals in an algebraic number field $K/\mathbb{Q}$

For each prime $p \in \mathbb{Z}$ let us choose exactly one prime above $\mathfrak{P} | (p) $ in $O_K$

Consider then $I'_K$ to be the subset of the fractional ideal group $I_K$ generated entirely by this subset of all the ideals in $K$

So if this is not wrong thinking my question is what would the group $I'_K/P'_K$ be ?