Closed formulas for the generators of all constacyclic codes and for the factorization of Xn − 1, the n-th cyclotomic polynomial and every composition of the form f(Xn) over a finite field for arbitrary positive integers n

Abstract

In this paper we present the solution to four 150 year-old closely related problems in the study of polynomials and algebraic codes over a finite field $F_q$. We give one closed formula each for the factorization of the polynomials $X^n-a$ for arbitrary $a\in F_q^{⁎}$, $X^n-1$, the n-th cyclotomic polynomial and the composition $f\left(X^n\right)$ for arbitrary monic irreducible polynomials $f\in F_q\left[X\right]$, $f\ne X$, for arbitrary positive integers n. With a new perspective on these problems we show that the factorization of $X^n-a$ has a beautiful underlying structure which is completely determined by the order of a in the multiplicative group $F_q^{⁎}$.

The binomial $X^n-a$ and the composition $f\left(X^n\right)$ were first considered over prime fields by Joseph Alfred Serret in 1866. In recent years, the factorization of $X^n-a$ has been studied extensively due to the fact that its factors are the generators of the popular constacyclic codes over finite fields. The factorization of $X^n-1$ and of its famous factor, the n-th cyclotomic polynomial, was first studied over prime fields by Carl Friedrich Gauss (among others) in the middle of the 19th century. Since then, many mathematicians were fascinated by this factorization and nowadays it is needed for the construction of the cyclic codes over finite fields.

Many formulas for the factorization of the four polynomials for special choices of n and a were obtained by a large number of mathematicians. Our formulas naturally extend, cover, combine and complete all of these partial solutions.