Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77-91, 1984] are here extended to differential polynomial rings over differential fields. Among the standard consequences of this work are: a partial answer to the primality problem, the equivalence of this problem with several others related to the Ritt problem, and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential Nullstellensatz.
We compute all the moments of a normalization of the function which counts unramified H 8 -extensions of quadratic fields, where H 8 is the quaternion group of order 8, and show that the values of this function determine a point mass distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for G in a large class of 2-groups, which we conjecture will give finite moments which determine a distribution. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified H 8 -extensions.
Let f (K) be the number of unramified extensions L/K of a quadratic number field K with Gal (L/K) = H and Gal (L/Q) = G where G is a central extension of F n 2 by F 2 . We find a function g (K) such that f /g has finite moments and a distribution on its values. We show this distribution is a point mass when H is non-abelian and the Cohen-Lenstra distribution when H is abelian, despite the fact that the set of values of f /g do not form a discrete set. We prove an explicit formula for f as well as a refined counting function with local conditions. We also determine correlations of such counting functions for different groups G. Lastly we formulate a conjecture about moments and correlations for any pair of 2-groups (G, H).
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