10.18453/rosdok_id00004733 Graner, Anna-Maurin Anna-Maurin Graner http://d-nb.info/gnd/1355286212 Universität Rostock 2024 510 Mathematics 2024 en https://purl.uni-rostock.de/rosdok/id00004733 urn:nbn:de:gbv:28-rosdok_id00004733-8 We present a closed explicit formula for all generating polynomials of a very popular set of codes - the constacyclic codes over finite fields. This problem is equivalent to the factorization of the polynomial X^n-a into monic irreducible factors over a finite field for all positive integers n. From our explicit formula for this factorization we also derive the factorization of the n-th cyclotomic polynomial and of any composition of the form f(X^n) where f is an irreducible polynomial. Additionally, we present a construction of a large set of irreducible polynomials of the same degree.