Properties shared by number fields with the same normal closure? - MathOverflow

Properties shared by number fields with the same normal closure?

2010-05-02T20:42:38Z

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?

Answer by Kevin Buzzard for Properties shared by number fields with the same normal closure?

2010-05-02T21:04:06Z

Fields with the same Galois closure can be completely different. For example take a random irreducible polynomial of huge degree, let $F$ be the field obtained by adjoining one root of the polynomial, and let $K$ be the field obtained by adjoining all of the roots. These fields have the same normal closure, and one is in the other, but really have very little to do with each other. There's no reason the class groups should coincide, they definitely don't have the same unit group in general or even the same unit group rank, etc etc. They are completely different animals. I think you're barking up the wrong tree.

On the other hand you can have two subfields of the complexes which aren't the same, but are abstractly isomorphic as fields (e.g. adjoining two different complex roots of one irreducible polynomial). And of course in this sense any invariant which is isomorphism-invariant (e.g. all the things you mentioned) will be the same for the two.

Answer by Franz Lemmermeyer for Properties shared by number fields with the same normal closure?

2010-05-03T04:29:19Z

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.