2 research outputs found
Closed formulas for the factorization of , the -th cyclotomic polynomial, and over a finite field for arbitrary positive integers
The factorizations of the polynomial and the cyclotomic polynomial
over a finite field have been studied for a very long
time. Explicit factorizations have been given for the case that
where , is prime or is the product of
two primes. For arbitrary the factorization of the
polynomial is needed for the construction of constacyclic codes. Its
factorization has been determined for the case and
for the case that there exist at most three distinct prime factors of and
for a prime . Both polynomials and
are compositions of the form for a monic irreducible
polynomial . The factorization of the composition
is known for the case and
for or prime.
However, there does not exist a closed formula for the explicit factorization
of either , the cyclotomic polynomial , the binomial or
the composition . Without loss of generality we can assume that
. Our main theorem, Theorem 18, is a closed formula for the
factorization of over for any and
any positive integer such that . From our main theorem we
derive one closed formula each for the factorization of and of the
-th cyclotomic polynomial for any positive integer such that
(Theorem 2.5 and Theorem 2.6). Furthermore, our main theorem
yields a closed formula for the factorization of the composition for
any irreducible polynomial , , and any positive
integer such that (Theorem 27).Comment: We added factorizations of and the -th cyclotomic
polynomial. We improved the selection of the parameters for our main theorem,
gave a more thorough proof for it and corrected the choice of the
representative system for the case . We included a reference
to [WY18]. In Proposition 6 we corrected the choice of for the case $a=1