Parameterizing solutions to any Galois embedding problem overZ/pnZwith elementary p -abelian kernel

Abstract

In this paper we use the Galois module structure for the classical parameterizing spaces for elementary p-abelian extensions of a field K to give necessary and sufficient conditions for the solvability of any embedding problem which is an extension of Z/p n Z with elementary p-abelian kernel. This allows us to count the total number of solutions to a given embedding problem when the appropriate modules are finite, and leads to some nontrivial automatic realization and realization multiplicity results for Galois groups.

$p$-groups have unbounded realization multiplicity

“…On the contrary, we reach our results by interpreting the appearance of certain Galois groups over a field F in terms of the Galois-module structure of well-known parameterizing spaces of elementary p-abelian extensions. This methodology has already been used to compute automatic realizations for certain groups (see [12,14]), and the second author of this paper extends the ideas developed in this paper to give module-theoretic interpretations for solutions to any embedding problems Ĝ → G → 1 for any group Ĝ which is an extension of G = Z/p n Z by an elementary p-abelian group in [15]. In particular this allows for a precise count of solutions to these embedding problems, as well as giving new automatic realization and realization multiplicity results.…”

Galois module structure of square power classes for biquadratic extensions

“…Generalizations to the case where K is characteristic p (but Gal(K/F ) is still assumed to be a cyclic p-group) have also been explored in [8,9,87,106] and make use of [62].…”

“…These stratified decompositions have, in turn, been translated into properties that distinguish absolute p-Galois groups within the larger class of pro-p groups. For example, using the structure of J(K), one is able to prove a variety of automatic realization results (i.e., the appearance of one group as a Galois group over K forces the appearance of some other group as a Galois group over K) and realization multiplicity results (i.e., computing or bounding the number of distinct extensions of K within K sep that have a prescribed Galois group) (see [8,27,86,90,106]). Results of this kind have been studied before, and sit naturally within the context of Galois embedding problems.…”

Quaternion algebras and square power classes over biquadratic extensions

“…One source of this interest comes from the observation that if K/F is a Galois extension with Gal(K/F ) = G, then the F p [G]-modules of K × /K ×p are in correspondence with elementary p-abelian extensions of K that are Galois over F . In fact, if L/K is the field extension corresponding to some F p [G]-submodule N, it is often possible to compute Gal(L/F ) strictly in terms of the module-theoretic structure of N, together with a small amount of field-theoretic information attached to the elements in N. The interested reader can consult [18] for some first results in this vein, and [17] for a generalization; in both of these cases, the group G is assumed to be a finite, cyclic p-group. In this way, it is typically possible to use power classes of a field as a parameterizing space for a much broader classes of Galois theoretic objects.…”

“…Since that time there has been a wealth of additional module decompositions. Most of these results have stuck to the case where Gal(K/F ) is a cyclic p group, and they include: the structure of K × /K ×p when Gal(K/F ) ≃ Z/p n Z in [12]; the structure of associated cohomology groups in [10,11]; and characteristic p-analogs of both of these objects in [2,3,17].…”