When calculating the ideal class group of a number field, it is common to start with the Minkowski bound, followed by decomposing finitely many prime ideals of norm less than that bound, and finding relations between these primes.
Is there a way of avoiding the use of Minkowski bound in the computation of the ideal class groups?
For example, could we use some exact-sequence (say) to show some isomorphism of the ideal class group with some other familiar group? Or maybe the Artin reciprocity isomorphism can aid us in this direction? Or even, per chance we can avail of some suitable resolutions for the computation of some cohomology groups?
As to why one wants to avoid the use of Minkowski bound at all, I just think that the idea of this bound is quite analytical, and there might be a way of algebraically calculating the ideal class group.
I googled and searched this site, but didn't find anything useful. The site I found that talks about the computation of the ideal class groups, either views the Minkowski bound as a fundamental ingredient of its arguments, or uses the class number formula for imaginary quadratic fields, which I would like to avoid as well.
Any hints, references, or ideas are welcomed, thanks in advance.
P.S. This it not to say that I want to avoid all results of the geometry or analysis, just that I want to know if there are any results in this direction.