Splitting Field Research Articles

On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial L n (−3−n)(x)=∑ j=0 n 1 j! (n−j+1)(n−j+2) 2 x j is irreducible over the rationals for all n⩾1 and has Galois group A n if n+1 is an odd square, and S n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L n ( α) ( x) can be embedded in a double cover.

On the number of equivalence classes of binary forms of given degree and given discriminant

We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We have proved more general results for binary forms with coefficients in the ring of S-integers of a number field.

Mössbauer Studies of Thermal Power Plant Coal and Fly Ash

Iron-57 Mossbauer spectroscopic studies were carried out at room temperature on samples of coal, slag (bottom ash) and mechanical ash collected from Bhatinda (India) thermal power plant. Hyperfine parameters such as isomer shift, quadrupole splitting and total internal magnetic field of 57Fe nuclei were used to characterize various iron-bearing minerals. The observed parameters indicate the presence of pyrite, siderite and ankerite in coal sample while magnetic fractions of mechanical ash and slag samples show the formation of hematite and Al-substituted magnesio-ferrite. The non-magnetic fraction of slag ash shows the dominance of Fe2+ phases while that of mechanical ash demonstrates the formation of both Fe2+ and Fe3+ phases. These findings are compared with Mossbauer and magnetic susceptibility studies on fly ash samples of Panipat (India) thermal power plant reported earlier.

A family of cyclic cubic polynomials whose roots are systems of fundamental units

We give a family of cyclic cubic polynomials whose roots are systems of fundamental units of the splitting fields. These polynomials are constructed by a linear fractional transformation from Shanks’ polynomials with rational coefficients.

Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dimes(X ) of X ,a s def ined by Oleg Izhboldin, is dimes(X ) = dim(X ) − i(X ) + 1 , where i(X ) is the first Witt index of X (i.e., the Witt index of X over its function field). Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X ). Our main theorem states that dimes(X ) ≤ dim(Y ) and that in the case dimes(X ) = dim(Y ) the quadric X is isotropic over F(Y ). Applying the main theorem to a projective quadric Y , we get a proof of Izhboldin's conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X ) ,t hen dim es(X ) ≤ dimes(Y ), and the equality holds if and only if X is isotropic over F(Y ). We also solve Knebusch's problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dimes(X ). Let (V ,ϕ ) be a non-degenerate quadratic form of dimension at least 2 over a field F of characteristic not 2 and let X = Q(ϕ) be the quadric hypersurface given by the equation ϕ(x) = 0 in the projective space P(V ). We say that the quadric X is anisotropic if ϕ is an anisotropic quadratic form. By Springer's theorem, every closed point of an anisotropic quadric X has

On stable equivalences of Morita type for finite dimensional algebras

In this paper, we assume that algebras are finite dimensional algebras with 1 over a fixed field $k$ and modules over an algebra are finitely generated left unitary modules. Let $A$ and $B$ be two algebras (where $k$ is a splitting field for $A$ and $B$) with no semisimple summands. If two bimodules $_{A}M_{B}$ and $_{B}N_{A}$ induce a stable equivalence of Morita type between $A$ and $B$, and if $N\otimes _{A}-$ maps any simple $A$-module to a simple $B$-module, then $N\otimes _{A}-$ is a Morita equivalence. This conclusion generalizes Linckelmann’s result for selfinjective algebras. Our proof here is based on the construction of almost split sequences.

On x 6 + x + a in Characteristic Three

Let F be a field of characteristic 3 and 0 ≠ a ∈ F. We show that the 10 ways to factor x6 + x + a into two cubics over the algebraic closure F are in natural Galois bijection with the 10 roots of x10 + ax + 1. We use this to (1) prove the two polynomials have the same splitting field; (2) prove that a difference set constructed by Arasu and Player using the polynomial x6 + x + a is isomorphic to a difference set constructed by Dillon using the polynomial x10 + x + a; (3) obtain a natural realization for the accidental isomorphism between the alternating group A6 and the special linear group PSL2(9); and (4) characterize how x6 + x + a factors when F e GF (3m) with m odd. For example, x6 + x + a is irreducible if and only if a can be written as δ− 36 + δ4 with δ ∈ F× and Tr(δ5) ≠ 0.

A study of Mssbauer spectroscopy of microcrystallization of amorphous Fe7 3.5Cu1Nb3Si13.5B9 irradiated b y a CO2-Laser

Microcrystalline phases were produced by laser irradiation on the Fe73.5Cu1Nb3Si13.5B9 amorphous ribbon with a power ranging from 40 to 170W,scaning speed 10mm/s,laser beam spot 20mm. Hyperfine structures of the original and crystallized samples were analyzed by transmission Mssbauer spectros copy. We discovered that the basic parameters of the Mssbauer spectra——isome r shift（IS）, quadrupole splitting（QS） and hyperfine magnetic field（Hhf） ch ange with the laser power. Experimental results show that the magnetic moment re organization of Fe73.5Cu1Nb3Si13.5 B9 amorphous phases has take n place and the atom magnetic moment direction maintains a preferred in-plane or ientation, The average hyperfine magnetic field of the amorphous phase is improv ed because of the microcrystalline phase. The crystallized phase was Fe-Si alloy with a DO3 structure, and the area of its spectral lines is about 2% —3.4%. This is caused by the jump of one electron in Fe atom 3d6 shell int o Si atom 3p2 shell, forming a stability electron group.

Broué's conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of N G ( D) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O'Nan simple group and the Higman–Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.

COCYCLE EQUIVALENT HOPF ALGEBRA ACTIONS

Let H be a Hopf algebra over a field K and assume a K-algebra A is an H-module algebra under two actions; · and ∘. We call these actions cocycle equivalent if there is an action of H on M 2(A), h ♦ X, such that for h ∈ H, X ∈ M 2(A) and a,b ∈ A. Two actions are cocycle equivalent if and only if there are cocycles that relate the two actions. Using these, it is shown that cocycle equivalence is a equivalence relation. Finally let H be a finite dimensional, semisimple, cocommutative Hopf algebra and assume K is a splitting field of H. It is shown that the Connes spectrum of H acting on M 2(A) is the intersection of the Connes spectra of H acting on A under · and ∘. Denote the smash product of A and H under the action · by (A#H,·). Let A be H-prime, then (A#H,·) is prime if and only if (A)#H, ♦) is prime if and only if (M 2(A)# H, ♦) is prime.

The extended version of the computer package CST for conversions, standardization and transformations of the spin Hamiltonian and the crystal-field Hamiltonian

The computer package CST (Conversions, Standardization and Transformations) is useful for general manipulations of the zero-field splitting (ZFS) and crystal field (CF) parameters for various systems, especially for transition ions at orthorhombic and lower symmetry sites in crystals. The ZFS parameters are extensively used in the EMR and related spectroscopic and magnetic studies (e.g. magnetic susceptibility, magnetic anisotropy, Mössbauer spectroscopy), whereas the CF parameters in optical absorption spectroscopy, inelastic neutron scattering, and infrared spectroscopy. An extended version of the CST package developed recently is presented in this paper. The standardization, including the standardization errors, for the five possible non-standard ranges of the ‘rhombicity’ parameter for monoclinic and triclinic symmetry in the local axis system has been worked out. The handling of the border points in the standardization module has been improved. The rotational invariants used before only for the transformations of the CF parameters have been incorporated into the standardization module for the CF and ZFS parameters expressed in any of the major tensor operator notations. The input option in the Euler angles has been added into the transformation module, and the newest physical constants in the unit conversions module have been adopted. Specific applications of the CST package for the CF and ZFS parameters will be presented elsewhere.

On a certain ideal of Külshammer in the centre of a group algebra

Let G be a finite group and let F be a splitting field of characteristic $ p > 0 $ . We show that I 2 = E 0, where I is a certain ideal of the centre Z of FG, and E 0 is the span of the block idempotents of defect zero.

Splitting fields of G-varieties

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is trivial. If $L/K$ is a (finite) Galois extension then $\Gal(L/K)$ is called a splitting group of $X$. We prove a lower bound on the size of a splitting field of $X$ in terms of fixed points of nontoral abelian subgroups of $G$. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.

GALOIS LATTICES AND REDUCTIONS OF ALGEBRAIC TORI

In this paper we construct the minimal integral model of a finite type of an algebraic torus defined over a complete non-Archimedean extension of an algebraic number field. The main problem is to study this model in the case of the ramified splitting field. We describe the models of quasi-split tori, norm tori and tori of low dimension. We also calculate the reduction of these models.

Kronecker's and Newton's Approaches to Solving: A First Comparison

These pages are a first attempt to compare the efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker's solution (from the symbolic approach) with approximate zero theory (introduced by S. Smale as a foundation of numerical analysis). For this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker solutions into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting field and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving.

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H=( a, b) F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over ( H, τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h K over K. This gives a map of Witt groups ρ:W −1( H, τ)→W( K) induced by ρ( h)= h K . When K is a generic splitting field of H we prove in this note that the map ρ is injective.

Electron coherent tunneling in low-dimensional magnetic quantum structures

We study electron coherent tunneling properties through the magnetic-barrier structure. The characteristics of wave-vector filtering, resonance splitting, and electric field effects have been investigated. The results indicate that the features of transmission resonance in the magnetic-barrier structure not only depend on the structure and the incident wave vector, but also strongly depend on the electric field. There exist rich and interesting features for electrons tunneling through the magnetic-barrier structure which is arranged with nonidentical magnetic barriers and wells.

On the Irreducible Representations of the Specializations in Characteristics 2 and 3 of the Generic Hecke Algebra of Type F4

A complete determination of the irreducible modules of specialized Hecke algebras of type F4, with respect to specializations with equal parameters, has been obtained by M. Geck and K. Lux (1991, Manuscripta Math.70, 285–306) for all characteristics. A similar determination for specializations with v=u2 and v=u4 has been obtained by K. Bremke (1994, Manuscripta Math.83, 331–346). In another paper (1999, J. Algebra, 218, 654–671), the authors determined the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3, obtaining en route decompositions of the generic irreducible modules under such specializations. In this paper, the corresponding results for characteristic 2 or 3 are obtained. Again, it is found that the decomposition matrices may be expressed in lower uni-triangular form in all these cases and that the splitting fields are those generated by the images of the parameters.

On iterates of Möbius transformations on fields

Let p be a quadratic polynomial over a splitting field K, and S be the set of zeros of p. We define an associative and commutative binary relation on G ≡ K ∪ {∞} - S so that every Mobius transformation with fixed point set S is of the form x plus c for some c. This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If K is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Mobius transformation m to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of m are distinct and the iterates of m converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.

Desingularization and integralityof the automorphism scheme of a finite field extension

Let $Aut_kK$ be the automorphism scheme of a finite purely inseparable field extension $k\to K$ (in fact, we consider a wide class of finite ring extensions). Let $A_M$ be the splitting algebra of K (see 2.8) and $K_M={A_M}_{\text red} $ the splitting field of K. It is proved that $Aut_kK$ is integral if and only if $A_M=K_M$ . This may be formulated as a condition on the degree of $K_M$ and generalizes a result of Chase. Surprisingly, $A_M$ may be defined intrinsically, since $\mathrm{Spec} A_M$ is the scheme parametrizing the maximal smooth subgroups of $Aut_kK$ . It is also proved that the desingularization of $Aut_kK$ is the universal maximal smooth subgroup of $Aut_kK$ and coincides with the blowing up along a closed subscheme canonically defined from the action of $Aut_kK$ on $A_M$ .