On the Lagrange Resolvents of a Dihedral Quintic Polynomial

The cyclic quartic field generated by the fifth powers of the Lagrange resolvents of a dihedral quintic polynomial f (x) is explicitly determined in terms of a generator for the quadratic subfield of the splitting field of f (x). Let f (x) = x 5 + px 3 + qx 2 + r x + s ∈ Q[x] be an irreducible quintic polynomial with a solvable Galois group. Let x 1 ,x 2 ,x 3 ,x 4 ,x 5 ∈ C be the roots of f (x). The splitting field of f is K = Q(x 1 ,x 2 ,x 3 ,x 4 ,x 5). Let ζ be a primitive fifth root of unity. The Lagrange resolvents of the root x 1 are r 1 = x 1 ,ζ = x 1 + x 2 ζ + x 3 ζ 2 + x 4 ζ 3 + x 5 ζ 4 ∈ K(ζ), r 2 = x 1 ,ζ 2

Let f (x) = x 5 + px 3 + qx 2 + r x + s ∈ Q[x] be an irreducible quintic polynomial with a solvable Galois group.Let x 1 ,x 2 ,x 3 ,x 4 ,x 5 ∈ C be the roots of f (x).The splitting field of f is K = Q(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ).Let ζ be a primitive fifth root of unity.The Lagrange resolvents of the root x 1 are We set By [1,Theorem 2] we know that the Galois group of f is Z 5 (cyclic group of order 5), D 5 (dihedral group of order 10), or F 20 (Frobenius group of order 20).When Gal(f ) D 5 , the splitting field K of f contains a unique quadratic subfield, say Q( √ m) (m squarefree integer ≠ 1).In this note we show, for quintic polynomials f with Gal(f ) D 5 , that the fields Q(R i ) (i = 1, 2, 3, 4) are the same cyclic quartic field and we give a simple explicit generator for this field.We prove the following theorem.
where Q( √ m) is the unique quadratic subfield of the splitting field K of f .
where l 0 ,l 1 ,l 2 ,l 3 ,l 4 ∈ K are given in [1, page 391] and satisfy As Gal(f ) D 5 , by [1,Theorem 2,page 397] the discriminant D of f is a square in Q.Thus, by [1, pages 392-397], l 1 , l 2 , l 3 , l 4 are the roots of a quartic polynomial belonging to Q[x], which factors over Q into two irreducible conjugate quadratics The roots of one of these quadratics (without loss of generality the first) are l 1 and l 4 , and the roots of the other are l 2 and l 3 .Thus where a, b, c, d ∈ Q, b ≠ 0 and d ≠ 0. Thus Next we define Hence, as 1 Similarly Using Maple we find that The roots of g(x) are (again using Maple) The quantities under the radicals are X + Y √ 5 and X − Y √ 5, where As the roots of g(x) belong to the cyclic quartic field as

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation