Generators for primary closures of Galois fields

AbstractWe continue to study the existence of (norm- and) trace-compatible sequences of primitive normal bases for prime power extensions of finite fields, introduced by the author in Hachenberger (Finite Fields Appl. 5 (1999) 378–385; in: D. Jungnickel, H. Niederreiter (Eds.), Proceedings of the Fifth International Conference on Finite Fields and Applications, Augsburg, August 1999, Springer, Heidelberg, 2001, pp. 208–223), and improve on some aspects of these papers

On the existence of primitive completely normal bases of finite fields

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, provided that $n=p^{\ell}m$ with $(m,p)=1$ and $q>m$

Further results on the Morgan-Mullen conjecture

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ for any $q$ and $n$. It is known that the conjecture holds for $n \leq q$. In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for $q\leq n\leq O(q^\epsilon)$, where $\epsilon=2$ for the asymptotic results and $\epsilon=1.25$ for the effective ones. For $n$ even we need to assume that $q-1\nmid n$.Comment: arXiv admin note: text overlap with arXiv:1709.0314

Primitive free cubics with specified norm and trace

The existence of a primitive free (normal) cubic x3 - ax2 + cx - b over a finite field F with arbitrary specified values of a (≠0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed

On the existence of some specific elements in finite fields of characteristic 2

AbstractLet q be a power of 2, n be a positive integer, and let Fqn be the finite field with qn elements. In this paper, we consider the existence of some specific elements in Fqn. The main results obtained in this paper are listed as follows:(1)There is an element ξ in Fqn such that both ξ and ξ+ξ−1 are primitive elements of Fqn if q=2s, and n is an odd number no less than 13 and s>4.(2)For q=2s, and any odd n, there is an element ξ in Fqn such that ξ is a primitive normal element and ξ+ξ−1 is a primitive element of Fqn if either n|(q−1), and n⩾33, or n∤(q−1), and n⩾30, s⩾6

Primitive normal pairs of elements with one prescribed trace

Let $q, n, m \in \mathbb{N}$ such that $q$ is a prime power, $m \geq 3$ and $a \in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($\alpha$, $f(\alpha)$) in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ such that Tr$_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(\alpha^{-1})=a$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with degree sum $n$. In particular, for $q=5^k, ~k \geq 5$ and degree sum $n=4$, we explicitly find at most 11 choices of $(q, m)$ where existence of such pairs is not guaranteed.Comment: 19 page