DIHEDRAL QUINTIC FIELDS WITH A POWER BASIS

@article{Lavallee2005DIHEDRALQF,
  title={DIHEDRAL QUINTIC FIELDS WITH A POWER BASIS},
  author={Melissa J. Lavallee and Blair K. Spearman and Kenneth S. Williams and Qiduan Yang},
  journal={Mathematical journal of Okayama University},
  year={2005},
  volume={47}
}
to be monogenic. Dummit and Kisilevsky[4] have shown that there exist infinitely many cyclic cubic fields whichare monogenic. The same has been shown for non-cyclic cubic fields, purequartic fields, bicyclic quartic fields, dihedral quartic fields by Spearman andWilliams [15], Funakura [6], Nakahara [14], Huard, Spearman and Williams[10] respectively. It is not known if there are infinitely many monogeniccyclic quartic fields. If 
View via Publisher
math.okayama-u.ac.jp
15 Citations
Highly Influential Citations
3
Background Citations
8
Methods Citations
5
Results Citations
1

Figures and Tables from this paper

15 Citations

Dihedral and cyclic extensions with large class numbers
This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups,…
On D(w)-quadruples in the rings of integers of certain pure number fields
The purpose of this paper is to show the non-existence of D(w)-quadruples in number fields of odd degree whose rings of integers are of the special form. We derive some elements which can not be…
A remark on the Lavallee-Spearman-Williams-Yang family of quadratic fields
In [4], M. J. Lavallee, B. K. Spearman, K. S. Williams and Q. Yang introduced a certain parametric D5-quintic polynomial and studied its splitting field. The present paper gives an infinite family of…
Relative Pólya group and Pólya dihedral extensions of Q
A note on families of monogenic number fields
We give a sufficient criterion for specializations of certain families of polynomials to yield monogenic number fields. This generalizes constructions in several earlier papers. As applications we…
On number fields with $k$-free discriminants
Given a finite transitive permutation group $G$, we investigate number fields $F/\mathbb{Q}$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This generalizes previous…
INTERSECTIVE POLYNOMIALS WITH GALOIS GROUP D 5
We give an infinite family of intersective polynomials with Galois group D5, the dihedral group of order 10. A monic polynomial f(x) with integer coefficients is called intersective if it has a root…
On number fields with power-free discriminant
Given a finite transitive permutation group G , we investigate number fields F/ℚ of Galois group G whose discriminant is only divisible by small prime powers. This generalizes previous investigations…
On number fields with $k$-free discriminants
Given a finite transitive permutation group G, we investigate number fields F/Q of Galois group G whose discriminant is only divisible by small prime powers. This generalizes previous investigations…
PSL(2, 5) sextic fields with a power basis
It is shown that there exist infinitely many PSLð2; 5Þ sextic fields with a power basis.
...
...

References

SHOWING 1-10 OF 36 REFERENCES
Integral Bases for Quartic Fields with Quadratic Subfields
Let L be a quartic number field with quadratic subfield Q([formula]). Then L = Q([formula], [formula]), where a + b[formula] is not a square in Q([formula]) and where a, b, and c may be taken to be…
ON A PROCEDURE FOR FINDING THE GALOIS GROUP OF A QUINTIC POLYNOMIAL
In [4, Proposition, pp. 883{884] a procedure is given to nd the Galois group of an irreducible quintic polynomial 2Z[x]. It is shown that this procedure does not always nd the Galois group.
Solving solvable quintics
5 3 2 Abstract. Let f{x) = x +px +qx +rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if…
Multiplicities of dihedral discriminants
Given the discriminant dk of a quadratic field k, the number of cyclic relative extensions N\k of fixed odd prime degree p with dihedral absolute Galois group of order 2p , which share a common…
The 2-Power Degree Subfields of the Splitting Fields of Polynomials with Frobenius Galois Groups
Abstract Let f(x) be an irreducible polynomial of odd degree n > 1 whose Galois group is a Frobenius group. We suppose that the Frobenius complement is a cyclic group of even order h. Let 2 t h. For…
Non monogénéité de l'anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5
Elementary and Analytic Theory of Algebraic Numbers
1. Dedekind Domains and Valuations.- 2. Algebraic Numbers and Integers.- 3. Units and Ideal Classes.- 4. Extensions.- 5. P-adic Fields.- 6. Applications of the Theory of P-adic Fields.- 7. Analytical…
Generic Polynomials: Constructive Aspects of the Inverse Galois Problem
This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is…
Cubic Fields with a Power Basis
It is shown that there exist infinitely many cubic fields L with a power basis such that the splitting field M of L contains a given quadratic field K.
Arithmetical Properties of Polynomials
1. Throughout this paper f (x) will denote a polynomial whose coefficients are integers with highest common factor 1, and 1 will denote the degree of f (x). We assume that the highest coefficient in…
...
...