15 research outputs found
Some notes on the -normal elements and -normal polynomials over finite fields
Recently, the -normal element over finite fields is defined and
characterized by Huczynska et al.. In this paper, the characterization of
-normal elements, by using to give a generalization of Schwartz's theorem,
which allows us to check if an element is a normal element, is obtained. In
what follows, in respect of the problem of existence of a primitive 1-normal
element in over , for all and , had
been stated by Huczynska et al., it is shown that, in general, this problem is
not satisfied. Finally, a recursive method for constructing -normal
polynomials of higher degree from a given -normal polynomial over
is given
On -normal elements over finite fields
The so called -normal elements appear in the literature as a
generalization of normal elements over finite fields. Recently, questions
concerning the construction of -normal elements and the existence of
-normal elements that are also primitive have attracted attention from many
authors. In this paper we give alternative constructions of -normal elements
and, in particular, we obtain a sieve inequality for the existence of
primitive, -normal elements. As an application, we show the existence of
primitive -normal elements for a significant proportion of 's in many
field extensions. In particular, we prove that there exist primitive
-normals in over in the case when lies
in the interval , has a special property and .Comment: 15 page
A note on additive characters of finite fields
Let be the finite field with elements, where is a prime
power and, for each integer , let be the unique
-degree extension of . The -orders of an element
in and an additive character over have been
extensively used in the proof of existence results over finite fields (e.g.,
the Primitive Normal Basis Theorem). In this note we provide an interesting
relation between these two objects
Variations of the Primitive Normal Basis Theorem
The celebrated Primitive Normal Basis Theorem states that for any
and any finite field , there exists an element that is simultaneously primitive and normal over . In
this paper, we prove some variations of this result, completing the proof of a
conjecture proposed by Anderson and Mullen (2014). Our results also imply the
existence of elements of with multiplicative order
and prescribed trace over .Comment: 19 page
The trace of 2-primitive elements of finite fields (amended version)
Let be a prime power and integers such that . An
element of of multiplicative order is called
\emph{-primitive}. For any odd prime power , we show that there exists a
-primitive element of with arbitrarily prescribed
trace when . Also we explicitly describe the values
that the trace of such elements may have when . A feature of this amended
version is the reduction of the discussion to extensions of prime degree .Comment: This is an amended version of [6
Existence of primitive -normal elements in finite fields
An element is \emph{normal} if forms a basis of as a vector space over ; in this case, is
a normal basis of over . The notion of
-normal elements was introduced in Huczynska et al (2013). Using the same
notation as before, is -normal if spans a
co-dimension subspace of . It can be shown that -normal
elements always exist in , and Huczynska et al (2013) show
that elements that are simultaneously primitive and -normal exist for and for large enough when (we note that primitive
-normals cannot exist when ). In this paper, we complete this theorem
and show that primitive, -normal elements of over exist for all prime powers and all integers , thus solving
Problem 6.3 from Huczynska, et al (2013).Comment: 29 page
The translate and line properties for 2-primitive elements in quadratic extensions
Let be integers and be any prime power such that . We say that the extension possesses the
line property for -primitive elements if, for every
, such that
, there exists some ,
such that has multiplicative order . Likewise,
if, in the above definition, is restricted to the value , we say
that possesses the translate property. In this
paper we take (so that necessarily is odd) and prove that
possesses the translate property for
2-primitive elements unless . With some additional
theoretical and computational effort, we show also that possesses the line property for 2-primitive elements unless .Comment: arXiv admin note: text overlap with arXiv:1906.08046,
arXiv:1903.0316
A new criterion on k-normal elements over finite fields
The notion of normal elements for finite fields extension has been
generalized as k-normal elements by Huczynska et al. [3]. The number of
k-normal elements for a fixed finite field extension has been calculated and
estimated [3], and several methods to construct k-normal elements have been
presented [1,3]. Several criteria on k-normal element have been given [1,2]. In
this paper we present a new criterion on k-normal elements by using idempotents
and show some examples. Such criterion has been given for usual normal element
before [6].Comment: 1
Finite field extensions with the line or translate property for -primitive elements
Let be integers and be any prime power such that . We say that the extension possesses the
line property for -primitive elements property if, for every
, such that
, there exists some ,
such that has multiplicative order . We prove
that, for sufficiently large prime powers ,
possesses the line property for -primitive elements. We also discuss the
(weaker) translate property for extensions.Comment: arXiv admin note: text overlap with arXiv:1903.0316
Character sums over affine spaces and applications
Given a finite field , a positive integer and an -affine space , we provide a new bound
on the sum , where a multiplicative
character of . We focus on the applicability of our estimate
to results regarding the existence of special primitive elements in . In particular, we obtain substantial improvements on previous works.Comment: Comments/suggestions are welcom