Notes on the existence of certain unramified 2-extensions

Nomura, Akito

doi:10.1215/ijm/1258138479

Cited by 5 publications

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“…With a little extra work, we can determine when a solution can be found with specific ramification. Some results of this nature in more specific instances were proven in [16] [17] [18] by Nomura and some general results can be found in [24]. Proof.…”

Section: The Unramified Brauer Embedding Problemmentioning

confidence: 88%

“…There are other results of a similar flavor to this one classifying unramified extensions of families of number fields. For instance, there are results for the ℓ-and ℓ 2torsion of the class group over cyclic degree ℓ fields [8] [9] [14] [15] [20] and some results for nonabelian p-extensions of quadratic fields or cyclic fields of prime degree q = p due to Nomura [16] [17] [18] [19]. Closest to the direction of this paper are results due to Lemmermeyer classifying unramified G-extensions of quadratic fields for G one of several small nonabelian 2-groups [11] [12], for example: Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations

Alberts¹

2017

Preprint

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We study solutions to the Brauer embedding problem with restricted ramification. Suppose G and A are a abelian groups, E is a central extension of G by A, and f : Gal(Q/Q) → G a continuous homomorphism. We determine conditions on the discriminant of f that are equivalent to the existence of an unramified lift f , i.e. a continuous homomorphism making the following diagram commute:As a consequence of this result, we use conditions on the discriminant of K for K/Q abelian to classify and count unramified nonabelian extensions L/K normal over Q where the (nontrivial) commutator subgroup of Gal(L/Q) is contained in its center. This generalizes a result due to Lemmermeyer, which states that a quadratic field Q( √ d) has an unramified extension normal over Q with Galois group H8 the quaternion group if and only if the discriminant factors d = d1d2d3 as a product of three coprime discriminants, at most one of which is negative, satisfying the following condition on Legendre symbols:for {i, j, k} = {1, 2, 3} and pi any prime dividing di.

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Section: The Unramified Brauer Embedding Problemmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations

Alberts¹

2017

Preprint

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show abstract

“…In the case when G is an abelian group, by class field theory, this problem is closely related to the ideal class group of k. Bachoc-Kwon [1] and Couture-Derhem [3] studied the case when k is a cyclic cubic field and G is the quaternion group of order 8. The author [10] studied the case when k is a cyclic quintic field and G is a certain non-abelian 2-group of order 32. For an odd prime p, let E 1 be the non-abelian group of order p 3 such that the exponent is equal to p. In [8], the author studied the case when k is a quadratic field and G = E 1 .…”

Section: Introductionmentioning

confidence: 99%