Some multiplicative equations in finite fields

45 - On some equations over finite fields par IOULIA BAOULINA

Arithmetic progressions in multiplicative groups of finite fields

Let G be a multiplicative subgroup of the prime field Fp of size |G| > p1−κ and r an arbitrarily fixed positive integer. Assuming κ = κ(r) > 0 and p large enough, it is shown that any proportional subset A ⊂ G contains non-trivial arithmetic progressions of length r. The main ingredient is the Szemerédi-Green-Tao theorem. Introduction. We denote by Fp the prime field with p elements and Fp its ...

Rep#2a: Finite subgroups of multiplicative groups of fields

This theorem generalizes the (well-known) fact that the multiplicative group of a finite field is cyclic. Most proofs of this fact can actually be used to prove Theorem 1 in all its generality, so there is not much need to provide another proof here. But yet, let us sketch a proof of Theorem 1 that requires only basic number theory. The downside is that it is very ugly. First, an easy number-th...

Some Special Equations in a Finite Field.

Solving equations in finite fields and some results concerning the structure of GF(pm)

Quasi-self-reciprocal polynomials (QSRP's), were introduced in [2] as a generalization of SRP's. They represent a superset of SRP's, since QSRP's are defined f*(x) = *f(x). This extends SRP's over GF (pk), p > 2, to include those for which f*(x) =-f(x). (15) Thus, if f(x) = x' +fr-ixr-' +f,-2x'-2 + a** +f,x + f,,, fnei =-fi, with f. =-1 since f(x) is manic. Polynomials defined by (15) are denot...