We present in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functionswhere α is the i-th real root of the polynomial P . These functions will allow us to represent and manipulate easily every continuous piecewise polynomial function through the use of the corresponding canonical form.It will be also shown how to produce a "rational" representation of each function Ci(P ) allowing its evaluation by performing only operations in Q and avoiding the use of any real algebraic number.
The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), andZassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers Q. In this article, we study some properties of the polynomials f ∈ Q[x] such that the Galois group Gal(f ) acts sharply k-transitively on its roots.
Dirichlet (J Reine Angew Math 24, 1842) announced a generalization of his class number formula for real quadratic fields to biquadratic fields containing Q(i ), by replacing the circular trigonometric functions with certain elliptic trigonometric functions. Subsequently, Nazimow (Ann Sci École Norm Sup (3) 5:147-176, 1888) published the corresponding formula. Here we analyze the Dirichlet-Nazimow formula and update its proof in the context of Class Field Theory.
We present a new characterization of dihedral Galois groups of rational irreducible polynomials. It allows us to reduce the problem of deciding whether the Galois group of an even degree polynomial is dihedral, and its computation in the affirmative case, to the case of a quartic or odd degree polynomial, for which algorithms already exist. The characterization and algorithm are extended to permutation groups of order 2n containing an n-cycle.
IntroductionGiven an irreducible polynomial f ∈ [ޑx], we consider the problem of deciding whether its Galois group is dihedral, and, if so, we compute a minimal set of generators with its explicit action on the set of roots.Methods are already known for polynomials of prime degree [Jensen and Yui 1982] and of odd degree [Williamson 1990]. Here we consider the case of even degree polynomials. For it, we provide a characterization of dihedral Galois groups, based on the behavior of f related to a quadratic subfield K of its splitting field and a certain prime number. The quadratic subfield must be determined in order to decide whether the Galois group is dihedral. In the affirmative case, the roots of f will be expressed as polynomials in a fixed root α and a primitive element of K over .ޑ For computing K , we propose to transform f , after certain reductions, into either a quartic or an odd degree polynomial whose splitting field contains K . Such reductions are made from the nontrivial central elements of the Galois group.In Section 2 we state the characterization of dihedral Galois groups, whereas Section 3 is devoted to the algorithm that decides whether the group is dihedral. Finally, in Section 4, we extend the results to groups of order 2n containing a cyclic subgroup of order n, taking advantage of their similarity to dihedral groups. MSC2000: primary 12Y05; secondary 11R32.
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