On a generalization of the Demǰanenko determinant

The Dem?anenko matrices are generalized with the averaged Bernoulli polynomials, and their determinants are computed by the Dedekind-Frobenius formula. This enables us to interpret geometrically the special values at negative integers of Dedekind zeta function of maximal real subfields as well as of imaginary subfields of cyclotomic fields. We utilize the structural properties of the group of reduced residue classes modm and those of (averaged) Bernoulli polynomials, thus appealing to the predominance of Galois theory over the fields themselves, which makes it possible to present very lucid reasoning of the whole picture.     

Arithmetical properties of finite graphs and polynomials

A study is made of asymptotic arithmetical properties of the isomorphism classes of certain types of finite graphs, and of certain polynomials over Galois fields.     

Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the smallest extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group. Received: 18 February 2002     

Recurrent Methods for Constructing Irreducible Polynomials over GF(2s)

The paper is devoted to some results concerning the constructive theory of the synthesis of irreducible polynomials over Galois fields GF(q), q=2^s. New methods for the construction of irreducible polynomials of higher degree over GF(q) from a given one are worked out. The complexity of calculations does not exceed O(n³) single operations, where n denotes the degree of the given irreducible polynomial. Furthermore, a recurrent method for constructing irreducible (including self-reciprocal) polynomials over finite fields of even characteristic is proposed.     

On the enumeration of polynomials with prescribed factorization pattern

We use generating functions over group rings to count polynomials over finite fields with the first few coefficients and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic n-smooth polynomials of degree m over a finite field, as well as the number of monic n-smooth polynomials of degree m with the prescribed trace coefficient.     

New generalized almost perfect nonlinear functions

APN (almost perfect non-linear) functions over finite fields of even characteristic are widely studied due to their applications to the design of symmetric ciphers resistant to differential attacks. This notion was recently generalized to GAPN (generalized APN) functions by Kuroda and Tsujie to odd characteristic p. They presented some constructions of GAPN functions, and other constructions were given by Zha et al. We present new constructions of GAPN functions both in the case of monomial and multinomial functions. Our monomial GAPN functions can be viewed as a further generalization of the Gold APN functions. We show that a certain technique used by Hou to construct permutations over finite fields also yields monomial GAPN functions. We also present several new constructions of GAPN functions which are sums of monomial GAPN functions, as well as new GAPN functions of degree p which can be written as the product of two powers of linearized polynomials. For this latter construction we describe some interesting differences between even and odd characteristic and also obtain a classification in certain cases.     

Additive polynomials for finite groups of Lie type

This paper provides a realization of all classical finite groups of Lie type as well as a number of exceptional ones (with low-dimensional representations) as Galois groups over function fields over F _q and derives explicit additive polynomials for the extensions. Our unified approach is based on results of Matzat which give bounds for Galois groups of Frobenius modules and uses the structure and representation theory of the corresponding connected linear algebraic groups.     

Riordan arrays and r-Stirling number identities

By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.     

Explicit construction of PSL(2,7) Fields over Q(t) Embeddable in SL(2,7) Fields

We employ the recent results of Mestre [C. R. Acad. Sei. Paris, t. 319, Série I, pp. 781-782] to exhibit polynomials over Q(t) with the property that their splitting fields are regular, have Galois group PSL(2,7), and are embeddable in fields with Galois group SL(2,7) over Q(t).     

Equations with absolute Galois group isomorphic to An

Criteria are given for polynomials of the type Xⁿ + aX³ + bX² + cX + d, to have Galois group over any finite number field isomorphic to A_n. We use them to construct, for every n, infinitely many polynomials with absolute Galois group isomorphic to A_n, covering so, the case n even, $\text{4 ? n}$, for which explicit equations were not known.     

Irreducible characters of 3′-degree of finite symmetric,general linear and unitary groups

Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of ${\mathbf{N}}_G(P)$, where P is a Sylow 3-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.     

On cubic Galois field extensions

We study Morton's characterization of cubic Galois extensions F/K by a generic polynomial depending on a single parameter s∈K. We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F. For K=Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.     

Some polynomials associated with the $${\vavec{}}$$-Whitney numbes

In the present article, we study three families of polynomials associated with the r-Whitney numbers of the second kind. They are the r-Dowling polynomials, r-Whitney–Fubini polynomials and the r-Eulerian–Fubini polynomials. Then we derive several combinatorial results by using algebraic arguments (Rota’s method), combinatorial arguments (set partitions) and asymptotic methods.     

On the Galois module structure over CM-fields

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extensionN of a givenCM-fieldK are invariant under the involution on the set of all realizations of Δ overK which is induced by complex conjugation onK and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number fieldK are completely characterized by ramification and Galois module structure.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Sharply k-Transitive Permutation Groups Viewed as Galois Groups

The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), and Zassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers \mathbbQ{\mathbb{Q}}. In this article, we study some properties of the polynomials f ? \mathbbQ[x]{f \in \mathbb{Q}[x]} such that the Galois group Gal(f) acts sharply k-transitively on its roots.     

On exceptional pq-groups

A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of F does not appear.In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q,which have unique subgroups of orders p and q.     

A parametric family of quintic polynomials with Galois group D5

We give a parametric family of quintic polynomials of the form x⁵ + ax + b (a, b ∈ $\text{Q}$) with dihedral Galois group D₅. Some properties of the fields defined by these polynomials are also described.     

The uniform convergence and precise asymptotics of generalized stochastic order statistics

Let X(i,n,m,k), i=1,…,n, be generalized order statistics based on F. For fixed r∈N, and a suitable counting process N(t), t>0, we mainly discuss the precise asymptotic of the generalized stochastic order statistics X(N(n)−r+1,N(n),m,k). It not only makes the results of Yan, Wang and Cheng [J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci. 26 (2) (2006) 237-244] as the special case of our result, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables.     

Cyclotomic factors of the descent set polynomial

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group Sn only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.