On completely free elements in finite fields

On Completely Free Elements in Finite Fields

On primitive elements in finite fields of low characteristic

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, Fqn , small q and large n, we show that the set of all low degree polynomials contains the expected number of primitive elements. The main theorem we prove is a bound for character sums over short intervals in function fields. Our result is unconditional and ...

Elements of High Order on Finite Fields from Elliptic Curves

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that the values on points of order small with respect to their degree of rational functions on an elliptic curve have high order. We discuss several special cases, including an old construction of Wiedemann, giving the first non-trivial estimate for the ...

Completely-Irreducible Elements

The following propositions are true: (1) For every sup-semilattice L and for all elements x, y of L holds ⌈⌉L(↑x∩ ↑y) = x ⊔ y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = x ⊓ y. (3) Let L be a non empty relational structure and x, y be elements of L. If x is maximal in (the carrier of L) \ ↑y, then ↑x \ {x} = ↑x ∩ ↑y. (4) Let L be a non empty relational structu...

Divergence-free finite elements on tetrahedral grids for k≥6

It was shown two decades ago that the Pk-Pk−1 mixed element on triangular grids, approximating the velocity by the continuous Pk piecewise polynomials and the pressure by the discontinuous Pk−1 piecewise polynomials, is stable for all k ≥ 4, provided the grids are free of a nearly-singular vertex. The problem with the method in 3D was posted then and remains open. The problem is solved partiall...