Pre-Pólya group in even dihedral extensions of ℚ

Investigating on Pólya groups [P. J. Cahen and J. L. Chabert Integer-Valued Polynomials, Mathematical Surveys and Monographs, Vol. 48 (American Mathematical Society, Providence, 1997)] in non-Galois number fields, Chabert [J. L. Chabert and E. Halberstadt, From Pólya fields to Pólya groups (II): Non-Galois number fields, J. Number Theory (2020), https://doi.org/10.1016/j.jnt.2020.06.008 ] introduced the notion of pre-Pólya group [Formula: see text], which is a generalization of the pre-Pólya condition, duo to Zantema [H. Zantema, Integer valued polynomials over a number field, Manuscripta Math. 40 (1982) 155–203]. In this paper, using class field theory, we describe the pre-Pólya group of a [Formula: see text]-field [Formula: see text], for [Formula: see text] an even integer, where [Formula: see text] denotes the dihedral group of order [Formula: see text]. Moreover, for special case [Formula: see text], we improve the Zantema’s upper bound on the maximum ramification in Pólya [Formula: see text]-fields.

CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.