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?



Properties shared by number fields with the same normal closure?
There's a huge amount of literature on this problem starting with 


* F. Gassmann,  *Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppen*, Math. Z. 25, 661-675 (1926)


Gassmann constructed number fields with the same normal closure in which almost all primes split in the same way. Later, the question morphed into "do zeta functions determine the number field", for example in


* B. de Smit, Bart; R. Perlis,
*Zeta functions do not determine class numbers*,
Bull. Am. Math. Soc., New Ser. 31, No.2, 213-215 (1994)


There's actually a whole book out there on this problem:


* N. Klingen, *Arithmetical similarities. Prime decomposition and finite 
group theory*, Oxford (1998)


where you can find the relevant literature up to the 1990s.


