While studying some class field theory there was a lot of talk on galois extensions. Of course. When talking about non-galois number fields, usually the text will quickly take the galois closure. At this point it occurred to me that this implies many properties are shared by number fields with the same galois closure.
So splitting of primes and ramification are controlled by the galois closure. But what other properties are shared?
Class group, unit group, higher cohomology and K-groups?