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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?

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What do you mean by higher cohomology? Since, if $K \subset L$ with $L$ Galois over $K$, there is an exact sequence $1 \rightarrow G_L \rightarrow G_K \rightarrow Gal(L/K) \rightarrow 1,$ one gets inflation-restriction sequences relating the cohomology of $G_K$ to the cohomology of $G_L$. How these behave will depend on the Galois module whose cohomology you are computing, and the nature of the group $Gal(L/K)$. –  Emerton May 3 '10 at 5:27
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2 Answers

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.

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I have to strongly disagree. For starters, I think you will find the following very interesting: "Class number and ramification in number fields", Armand Brumer and Michael Rosen, Nagoya Math. J. Volume 23 (1963), 97-101. –  Dror Speiser May 2 '10 at 22:33
    
Of course there are also many results on units which I am sure you are aware of. For example for CM fields there is the relationship between the units of the maximal real subfield and the whole. –  Dror Speiser May 2 '10 at 22:43
    
The different (from closure down) are also usually very similar in a sense, specifically their radical. So data that isn't shared in class groups might well be shared in regulator. –  Dror Speiser May 2 '10 at 22:52
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I vote with Kevin. For coarse invariants like which primes ramify (which is radical of different in other words) one sees no difference, but you mention much finer invariants such as class group, unit group, cohomology, etc. It is really hard to predict what the degree of a Galois closure will be, let alone the structure of such refined invariants. The CM example is not relevant, since a totally real extension of a number field cannot ever have Galois closure equal to a quadratic CM extension. By the way, according to Math Reviews, the main result of the Brumer/Rosen paper is wrong. –  BCnrd May 3 '10 at 0:50
    
Dror, here is a way to create examples of non-isomorphic number fields with the same Galois closure: find a finite group G and nonconjugate subgroups H_1 and H_2 such that the only normal subgroup of G inside either one is trivial. If you can realize G as a Galois group over Q, the fields fixed by H_1 and H_2 are number fields with the same Galois closure and you can explore how much they have in common. As an explicit example, use G = A_4 with H_1 = <(123)> and H_2 = <(12)(34)>. The first has fixed field of degree 4 over Q and the second has fixed field that's of degree 6 over Q. –  KConrad May 3 '10 at 5:25
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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.

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Aah but this is a different and much more interesting question, in some sense. The original question was "which subtle properties of field X are inherited by field Y if X and Y have the same Galois closure". You are answering the question "find interesting examples of number fields X and Y which are not isomorphic but which are arithmetically similar". You're saying such examples exist. Dror was asking whether every example was interesting, in some sense. –  Kevin Buzzard May 3 '10 at 8:05
    
Perhaps I should have been more specific. But Perlis and de Smit have found number fields with the same Galois closure and different class numbers, as well as bounds on how much they can differ. For fields with the same zeta function, this also answers the question concerning the regulator. It is true, though, that I am only looking at a subset of the examples Dror has had in mind. –  Franz Lemmermeyer May 4 '10 at 5:56
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