40,725 research outputs found
On the number of -free elements with prescribed trace
In this paper we derive a formula for the number of -free elements over a
finite field with prescribed trace, in particular trace zero, in
terms of Gaussian periods. As a consequence, we derive a simple explicit
formula for the number of primitive elements, in quartic extensions of Mersenne
prime fields, having absolute trace zero. We also give a simple formula in the
case when is prime. More generally, for a positive integer
whose prime factors divide and satisfy the so called semi-primitive
condition, we give an explicit formula for the number of -free elements with
arbitrary trace. In addition we show that if all the prime factors of
divide , then the number of primitive elements in , with
prescribed non-zero trace, is uniformly distributed. Finally we explore the
related number, , of elements in with
multiplicative order and having trace . Let such that , where is the largest factor of
with the same radical as that of . We show there exists an element in
of (large) order with trace if and only if and . Moreover we derive an explicit formula for the
number of elements in with the corresponding large order
and having absolute trace zero, where is a Mersenne
prime
Primitive normal pairs of elements with one prescribed trace
Let such that is a prime power, and
. We establish a sufficient condition for the existence of a
primitive normal pair (, ) in over
such that
Tr, where is a rational function with degree sum . In particular,
for and degree sum , we explicitly find at most 11
choices of where existence of such pairs is not guaranteed.Comment: 19 page
Primitive elements in finite fields with arbitrary trace
Arithmetic of finite fields is not only important for other branches of mathematics but also widely used in applications such as coding and cryptography. A primitive element of a finite field is of particular interest since it enables one to represent all other elements of the field. Therefore an extensive research has been done on primitive elements, especially those satisfying extra conditions. We are interested in the existence of primitive elements in extensions of finite fields with prescribed trace value. This existence problem can be settled by means of two important theories. One is character sums and the other is the theory of algebraic function fields. The aim of this thesis is to introduce some important properties of these two topics and to show how they are used in answering the existence problem mentioned above
Primitive polynomials with prescribed second coefficient
The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite fieldwith any coefficient arbitrarily prescribed. This has recently been provedwhenever n ≥ 9. It is also known to be truewhen n ≤ 3.We showthat there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of xn−2) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x4 + a1x3 + x2 + a3x + 1 over the binary field. For n ≥ 6 we prove a stronger result, namely that the primitive polynomialmay also have its constant termprescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs
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
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