Dihedral Extensions Research Articles

An improved error term for counting D4$D_4$‐quartic fields

AbstractWe prove that the number of quartic fields with discriminant whose Galois closure is equals , improving the error term in a well‐known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.

Rationality problem for norm one tori for dihedral extensions

We give a complete answer to the rationality problem (up to stable k-equivalence) for norm one tori RK/k(1)(Gm) of K/k whose Galois closures L/k are dihedral extensions with the aid of Endo and Miyata [17, Theorem 1.5, Theorem 2.3] and Endo [15, Theorem 2.1]. By using a similar technique, we give refinements of the proof of stably rational cases of Endo and Miyata's theorems as an appendix of the paper.

On [formula omitted]-Hopf-Galois structures

Let K/F be a finite Galois extension of fields with Gal(K/F)=Γ. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. The dihedral group is one particular example of a semidirect product of Zn and Z2. In this article we count the number of Hopf-Galois structures with Galois group Γ of type G, where Γ,G are groups of the form Zn⋊ϕZ2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.

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.

Genus numbers of cyclic and dihedral extensions of prime degree

We study genus numbers of cyclic and dihedral number fields of prime degree $l\geq 5$. For cyclic number fields, we obtain definitive results. For the dihedral case, by assuming a conjecture on the average of $l$-part class numbers, we obtain partial resu

Relative Pólya group and Pólya dihedral extensions of [formula omitted

A number field with trivial Pólya group [2] is called a Pólya field. We define “relative Pólya group Po(L/K)” for L/K a finite extension of number fields, generalizing the Pólya group. Using cohomological tools in [1], we compute some relative Pólya groups. As a consequence, we generalize Leriche's results in [17] and prove the triviality of relative Pólya group for the Hilbert class field of K. Then we generalize our previous results [19] on Pólya S3-extensions of Q to dihedral extensions of Q of order 2l, for l an odd prime. We also improve Leriche's upper bound in [16] on the number of ramified primes in Pólya Dl-extensions of Q and prove that for a real (resp. imaginary) Pólya Dl-extension of Q at most 4 (resp. 2) primes ramify.

CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS

AbstractWe 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.

Monogenic dihedral quartic extensions

In this paper, we prove that the number of monogenic dihedral quartic extensions of absolute discriminants \(\le X\) is of size \(O(X^{\frac{3}{4}}(\log X)^3)\).

Comparing local constants of ordinary elliptic curves in dihedral extensions

In this paper, we study the theories of analytic and arithmetic local constants of elliptic curves, with the work of Rohrlich, for the former, and the work of Mazur and Rubin, for the latter, as a basis. With the Parity Conjecture as motivation, one expects that the arithmetic local constants should be the algebraic additive counterparts to ratios of local analytic root numbers. We calculate the constants on both sides in various cases, establishing this connection for a substantial class of elliptic curves. By calculating the arithmetic constants in some new cases, we also extend the class of elliptic curves for which one can determine lower bounds for the growth of p-Selmer rank in dihedral extensions of number fields.

Rédei's Triple Symbols and Modular Forms

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol $[a_1,a_2, p]$ describes the decomposition law of a prime number $p$ in a certain dihedral extension over $\mathbb{Q}$ of degree 8 determined by $a_1$ and $a_2$. In this paper, we show that the triple symbol $[-p_1,p_2, p_3]$ for certain prime numbers $p_1, p_2$ and $p_3$ can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.

Constructing and tabulating dihedral function fields

We present algorithms for constructing and tabulating degree-‘ dihedral extensions of Fq.x/, where q 1 mod 2‘. We begin with a Kummer-theoretic algorithm for constructing these function fields with prescribed ramification and fixed quadratic resolvent field. This algorithm is based on the proof of our main theorem, which gives an exact count for such fields. We then use this construction method in a tabulation algorithm to construct all degree-‘ dihedral extensions of Fq.x/ up to a given discriminant bound, and we present tabulation data. We also give a formula for the number of degree-‘ dihedral extensions of Fq.x/ with discriminant divisor of degree 2.‘ 1/, the minimum possible.

Transcendental values of class group $L$-functions, II

Let K be an imaginary quadratic field and f an integral ideal. Denote by Cl(f) the ray class group of f. For every non-trivial character χ of Cl(f), we show that L(1, χ)/π is transcendental. If f = f, then complex conjugation acts on the character group of Cl(f). Denoting by Ĉl(f) + the orbits of the group of characters, we show that the values L(1, χ) as χ ranges over elements of Ĉl(f) + are linearly independent over Q. We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin L-series attached to dihedral extensions.

On Brauer–Kuroda type relations of S-class numbers in dihedral extensions

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer–Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D2q and for arbitrary Galois-stable sets of primes S, containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D2q-extension from class numbers and S-class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.

The Brauer–Kuroda formula for higher S-class numbers in dihedral extensions of number fields

Let p be an odd prime and let L/k be a Galois extension of number fields whose Galois group is isomorphic to the dihedral group of order 2p. Let S be a finite set of primes of L which is stable under the action of Gal(L/k). The Lichtenbaum conjecture on special values of the Dedekind zeta function at negative integers, together with Brauer formalism for Artin's L-functions, gives a (conjectural) formula relating orders of motivic cohomology groups of rings of S-integers and higher regulators of S-units of the subextensions of L/k. In analogy with the classical case of special values at 0, we give an algebraic proof of this formula, i.e. without using the Lichtenbaum conjecture nor Brauer formalism. Our method also gives an interpretation of the regulator term as a higher unit index.

Odd parts of tame kernels of dihedral extensions

Odd parts of tame kernels of dihedral extensions

Decomposition of places in dihedral and cyclic quintic trinomial extensions of global fields

In this paper, we give a complete and explicit description of the splitting behavior of any place in a quintic trinomial dihedral or cyclic extension of a rational function field of finite characteristic distinct from 2 and 5. Our characterization depends only on the order of the base field and a parametrization of the coefficients of the generating trinomial. Moreover, we contrast some of our results to trinomial dihedral number fields of prime degree, where the unit rank behaves quite differently from the function field scenario.

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.

Galois extensions ramified only at one prime

This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over Q and over F q ( t ) ramified only at one finite prime. Over Q , we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over F q ( t ) restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over F q ( t ) ramified only at one prime.

Growth of Selmer rank in nonabelian extensions of number fields

Let p be an odd prime number, let E be an elliptic curve over a number field k, and let F/k be a Galois extension of degree twice a power of p. We study the Zp-corank rkp(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rkp(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, if G=Gal(F/K), if G+ is the subgroup of elements of G commuting with a choice of involution of F over k, and if rkp(E/K) is odd, then we show that (under mild hypotheses) rkp(E/F)≥[G:G+]. As a very specific example of this, suppose that A is an elliptic curve over Q with a rational torsion point of order p and without complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p such that every prime where both E and A have bad reduction has odd order in Fp× and such that the negative of the conductor of E is not a square modulo p, then there is a positive constant B depending on A but not on E or n such that rkp(E/Q(A[pn]))≥Bp2n for every n

The Mahler measure of dihedral extensions

The Mahler measure of dihedral extensions