Cyclotomy and permutation polynomials of large indices

We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices

Cyclotomy and analytic geometry over F_1

Geometry over non--existent "field with one element" $F_1$ conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of "analytic functions" over $F_1$. The paper combines a focused survey with some new constructions. In new version, several local additions and changes are made, references added.Comment: 30 page