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 

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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
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
...
...