Let q be a prime power, Fq be the finite field of order q and Fq(x) be the field of rational functions over Fq. In this paper we classify and count all rational functions ϕ ∈ Fq(x) of degree 3 that induce a permutation of P 1 (Fq). As a consequence of our classification, we can show that there is no complete permutation rational function of degree 3 unless 3 | q and ϕ is a polynomial.

Abstract. Let K be a number field with ring of integers O. After introducing a suitable notion of density for subsets of O, generalizing that of natural density for subsets of Z, we show that the density of the set of coprime m-tuples of algebraic integers is 1/ζK (m), where ζK is the Dedekind zeta function of K. This generalizes a result found independently by Mertens (1874) and Cesàro (1883) concerning the density of coprime pairs in Z.

For a set S of quadratic polynomials over a finite field, let C be the (infinite) set of arbitrary compositions of elements in S . In this paper we show that there are examples with arbitrarily large S such that every polynomial in C is irreducible. As a second result, when \#S > 1 , we give an algorithm to determine whether all the elements in C are irreducible, using only O( \#S(\log q)^3 q^{1/2} ) operations.

We provide a Galois theoretical framework which allows to produce good polynomials for the Tamo and Barg construction of optimal locally recoverable codes (LRC). Our approach allows to prove existence results and to construct new good polynomials, which in turn allows to build new LRCs. The existing theory of good polynomials fits in our new framework.2010 Mathematics Subject Classification. 11T06.

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