My experience with algebraic numbery theory has been as follows:
I studied about number fields, extensions $Q$. That required dealing with Dedekind domains and proving various results such as unique factorziation, $\sum e_i d_i = [K:Q]$, inertial group, etc.
I am now studying about the algebraic theory of local fields where everything is much simpler (including all the claims above, with appropriate definitions).
I have yet to see a major result one can deduce about number fields via the local fields (I understand that class field will do), I would like to find a way to avoid having to prove the results in number fields that I mentioned above, and find a way to deduce them via the local methods.
Sadly the current method I'm aware of to connect a number field to an extension of $Q_p$ requires already knowing about unique factorization and the fractional ideals, you consider the evaluation by some prime over $p$ to embed your field in an extension of $Q_p$.
Is there a way to develop ANT from local fields?