Unramified Subextensions of Ray Class Field Towers

Hajir, Farshid; Maire, Christian

doi:10.1006/jabr.2001.9079

Cited by 10 publications

(7 citation statements)

References 22 publications

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“…However, e.g., it is not known whether for a given number field with infinite p-class field tower k |k, there exists a finite subextension K|k such that Gal(k |K) satisfies the Golod-Šafarevič inequality (cf. [7,8] [6] predicts that for the Iwasawa μ-and λ-invariants we have…”

Section: Remarkmentioning

confidence: 93%

On $p$-class groups and the Fontaine-Mazur conjecture

Gärtner

2014

Mathematical Research Letters

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Abstract. Answering a question by Stark, we show that for an infinite unramified pro-p-extension of a number field k, the p-class numbers of its finite subextensions tend to infinity. This is proven by means of a group-theoretical result on compact p-adic analytic groups. Furthermore, we provide an equivalent formulation of the FontaineMazur conjecture for p-extensions unramified outside a finite set of primes not containing any prime above p.

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Section: Remarkmentioning

confidence: 93%

On $p$-class groups and the Fontaine-Mazur conjecture

Gärtner

2014

Mathematical Research Letters

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“…Therefore the assumptions of Theorem 4.4 are fulfilled. Similarly, the group P Sp 4 (3).2 has a rationally rigid triple of classes of element orders (2,8,9), and the classes of elements of order 8 and 9 respectively fulfill the required assumptions. Finally, the simple group P Sp 6 (2) has a rationally rigid triple of classes of element orders (2,7,9), and the classes of elements of order 7 and 9 respectively fulfill the required assumptions.…”

Section: General Criteriamentioning

confidence: 99%

“…Abhyankar's lemma then implies that L := L 0 M is an unramified extension of M with Gal(L/M) ∼ = G. Moreover, the resulting extension L/M is tamely defined over K, that is, there is a tame Galois extension L 0 /K such that L 0 ⊗ K M ∼ = L. This is a common method for generating unramified extensions, e.g. used in [15], [18], [9], [14], [24]. In fact, since the Inverse Galois Problem is known for many groups G, we restrict our consideration to such groups and hence choose K = Q.…”

Section: Introductionmentioning

confidence: 99%

Unramified extensions over low degree number fields

König

Neftin

Sonn

2020

Journal of Number Theory

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For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.On the other hand, It is well known that every finite group G appears as a Galois group of an unramified extension over some number field. This is obtained by realizing G as a Galois group of a tame (tamely ramified) extension L 0 /K over some number field, and finding a (not necessarily Galois) number field M which is disjoint from L 0 and satisfies: "for every prime P of M, the ramification index of P over its restriction p to K is divisible by the ramification index of p in L 0 ". Abhyankar's lemma then implies that L := L 0 M is an unramified extension of M with Gal(L/M) ∼ = G. Moreover, the resulting extension L/M is tamely defined

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“…Schmitals [11] and Schoof [12] produced a few isolated examples of this type. See also [3], [7], etc. For p P t2, 3, 5u, Hoelscher [4] announced the existence of number fields unramified outside tp, 8u and having an infinite Hilbert class field tower.…”

mentioning

confidence: 99%