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### An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

Open Mathematics

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L p,k, and the finite field F(p k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ1 of the variety V(L p,k) generated by L p,k into the variety V(F(p k)) generated by F(p k) and an interpretation Φ2 of V(F(p k)) into V(L p,k) such that Φ2Φ1(B) = B for every B ε V(L p,k) and Φ1Φ2(R) = R for every R ε V(F(p k)).

### An explicit description of the set of all normal bases generators of a finite field

Czechoslovak Mathematical Journal

Acta Arithmetica

### Automorphisms of completely primary finite rings of characteristic p

Colloquium Mathematicae

A completely primary ring is a ring R with identity 1 ≠ 0 whose subset of zero-divisors forms the unique maximal ideal . We determine the structure of the group of automorphisms Aut(R) of a completely primary finite ring R of characteristic p, such that if is the Jacobson radical of R, then ³ = (0), ² ≠ (0), the annihilator of coincides with ² and $R/\cong GF\left({p}^{r}\right)$, the finite field of ${p}^{r}$ elements, for any prime p and any positive integer r.

### Chebyshev polynomials and Pell equations over finite fields

Czechoslovak Mathematical Journal

We shall describe how to construct a fundamental solution for the Pell equation ${x}^{2}-m{y}^{2}=1$ over finite fields of characteristic $p\ne 2$. Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation ${x}^{2}-m{y}^{2}=n$.

### Construction of the mutually orthogonal extraordinary supersquares

Open Mathematics

Our purpose is to determine the complete set of mutually orthogonal squares of order d, which are not necessary Latin. In this article, we introduce the concept of supersquare of order d, which is defined with the help of its generating subgroup in ${𝔽}_{d}×{𝔽}_{d}$ . We present a method of construction of the mutually orthogonal supersquares. Further, we investigate the orthogonality of extraordinary supersquares, a special family of squares, whose generating subgroups are extraordinary. The extraordinary subgroups...

### Corps fini ou champ de Galois

Séminaire Dubreil. Algèbre et théorie des nombres

Acta Arithmetica

### Lower bounds on the class number of algebraic function fields defined over any finite field

Journal de Théorie des Nombres de Bordeaux

We give lower bounds on the number of effective divisors of degree $\le g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.

### On canonical subfield preserving polynomials

Acta Arithmetica

Explicit monoid structure is provided for the class of canonical subfield preserving polynomials over finite fields. Some classical results and asymptotic estimates will follow as corollaries.

### On the number of zero trace elements in polynomial bases for F2n.

Revista Matemática Complutense

Let Fq denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F2n over F2. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.

### On the set of distances between two sets over finite fields.

International Journal of Mathematics and Mathematical Sciences

### Orthogonality and complementation in the lattice of subspaces of a finite vector space

Mathematica Bohemica

We investigate the lattice $𝐋\left(𝐕\right)$ of subspaces of an $m$-dimensional vector space $𝐕$ over a finite field $\mathrm{GF}\left(q\right)$ with a prime power $q={p}^{n}$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $𝐋\left(𝐕\right)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $𝐋\left(𝐕\right)$ is orthomodular. For...

### Quantitative sum product estimates on different sets.

The Electronic Journal of Combinatorics [electronic only]

### The radical factors of $f\left(x\right)-f\left(y\right)$ over finite fields.

International Journal of Mathematics and Mathematical Sciences

Acta Arithmetica

Algebra i Logika

Algebra i Logika

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