9,067 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
Invariant tensors for simple groups
The forms of the invariant primitive tensors for the simple Lie algebras A_l,
B_l, C_l and D_l are investigated. A new family of symmetric invariant tensors
is introduced using the non-trivial cocycles for the Lie algebra cohomology.
For the A_l algebra it is explicitly shown that the generic forms of these
tensors become zero except for the l primitive ones and that they give rise to
the l primitive Casimir operators. Some recurrence and duality relations are
given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3)
and su(4) are also provided. Finally, new relations involving the d and f su(n)
tensors are given.Comment: Latex file. 34 pages. (Trivial) misprints corrected. To appear in
Nucl. Phys.
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
Fast Arithmetics in Artin-Schreier Towers over Finite Fields
An Artin-Schreier tower over the finite field F_p is a tower of field
extensions generated by polynomials of the form X^p - X - a. Following Cantor
and Couveignes, we give algorithms with quasi-linear time complexity for
arithmetic operations in such towers. As an application, we present an
implementation of Couveignes' algorithm for computing isogenies between
elliptic curves using the p-torsion.Comment: 28 pages, 4 figures, 3 tables, uses mathdots.sty, yjsco.sty Submitted
to J. Symb. Compu
p-Adic estimates of Hamming weights in Abelian codes over Galois rings
A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more
- …