A note on powers in finite fields

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years, the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection, it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation that is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.     

A note on Pall partitions over finite fields

A Pall partition for a quadratic space V is a collection of disjoint (except for {0}) maximal totally isotropic subspaces whose union contains all of the isotropic vectors in V. In this paper it is shown that no non-degenerate quadratic space of dimension 4k+1, k?1, over a finite field of odd characteristic can have a Pall partition. The method of proof consists of assuming such a partition exists and showing by various counting arguments that this leads to the existence of an impossible array of ordered pairs.     

A note on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form $x+\gamma {\mathrm{Tr}}_{q^n/q}(x^k)$ is also presented, which generalizes two examples of [10].     

Weakly o-minimal expansions of ordered fields of finite transcendence degree

Using a work of Diaz concerning algebraic independence of certainsequences of numbers, we prove that if K is a field of finitetranscendence degree over the rationals, then every weakly o-minimalexpansion of (K,,+,·) is polynomially bounded. In thespecial case where K is the field of all real algebraic numbers,we give a proof which makes use of a much weaker result fromtranscendental number theory, namely, the Gelfond–Schneidertheorem. Apart from this we make a couple of observations concerningweakly o-minimal expansions of arbitrary ordered fields of finitetranscendence degree over the rationals. The strongest resultwe prove says that if K is a field of finite transcendence degreeover the rationals, then all weakly o-minimal non-valuationalexpansions of (K,,+,·) are power bounded.     

A note on isometries over local or finite fields

Let K   be a finite or a local field of characteristic ≠2$\ne 2$. We give a new proof, in a slightly more general case, for the following classical theorem of Milnor. If two unitary operators of a quadratic space over K have the same irreducible minimal polynomial, then they are conjugate via a unitary operator. Our arguments are short and elementary.     

A note on the transcendence of infinite products

The paper deals with several criteria for the transcendence of infinite products of the form $\prod\limits_{n = 1}^\infty {[{b_n}{a^{{a_n}}}]/{b_n}{a^{{a_n}}}} $ where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a _n } _n=1 ^∞ and {b _n } _n=1 ^∞ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P.Corvaja, U.Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mend`es France, Acta Math. 193, (2004), 175–191).     

A note on composed products of polynomials over finite fields

Brawley and Carlitz introduced the method of composed products in order to construct irreducible polynomials of large degree from polynomials of lower degree. A basic ingredient of their construction is a binary operation on a subset \(G \subseteq {\bar{\mathbb{F }}_{q}}\) having certain properties. In this paper we classify all such binary operations when \(|G|= \infty \) (which is the most interesting case) and show that field addition and field multiplication are essentially the only such operations.     

A correspondence of certain irreducible polynomials over finite fields

An explicit correspondence between certain cubic irreducible polynomials over ${\mathbb{F}}_q$ and cubic irreducible polynomials of special type over ${\mathbb{F}}_{q^2}$ was established by Kim et al. In this paper, we give a generalization of their result to irreducible polynomials of odd prime degree. Our result includes the result of Kim et al. as a special case where the degree is three.     

A note on adjacency preservers on hermitian matrices over finite fields

It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form.     

A note on l-class groups of certain algebraic number fields

The structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number fields whose Galois groups are of type (p, p); (ii) non-Galois extensions $\text{Q(p}\text{d}{}_0\text{3}\text{,p}\text{d}{}_1\text{3}\text{)}$ of degree p² over Q; (iii) dihedral extensions of degree 2^{n + 1} over Q. It is shown that it is possible to obtain class number relations by group-theoretic methods. Subgroups of ideal class groups whose orders are prime to the extension degree are considered.     

A note on inverses of cyclotomic mapping permutation polynomials over finite fields

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.     

Uniqueness of certain Fourier-Jacobi models over finite fields

In this paper, we prove the uniqueness of certain Fourier-Jacobi models for the split exceptional group ${\mathrm{G}}_2$ over finite fields with odd characteristic. Similar results are also proved for ${\mathrm{Sp}}_4$ and ${\mathrm{U}}_4$.