7 research outputs found
Sequences of binary irreducible polynomials
In this paper we construct an infinite sequence of binary irreducible
polynomials starting from any irreducible polynomial f_0 \in \F_2 [x]. If
is of degree , where is odd and is a
non-negative integer, after an initial finite sequence of polynomials with , the degree of is twice the degree
of for any .Comment: 7 pages, minor adjustment
Sequences of irreducible polynomials without prescribed coefficients over odd prime fields
In this paper we construct infinite sequences of monic irreducible
polynomials with coefficients in odd prime fields by means of a transformation
introduced by Cohen in 1992. We make no assumptions on the coefficients of the
first polynomial of the sequence, which belongs to \F_p [x], for some
odd prime , and has positive degree . If for
some odd integer and non-negative integer , then, after an initial
segment with , the degree of the polynomial
is twice the degree of for any .Comment: 10 pages. Fixed a typo in the reference
Construction of irreducible polynomials through rational transformations
Let be the finite field with elements, where is a power
of a prime. We discuss recursive methods for constructing irreducible
polynomials over of high degree using rational transformations.
In particular, given a divisor of and an irreducible polynomial
of degree such that is even or , we show how to obtain from a sequence of
irreducible polynomials over with .Comment: 21 pages; comments are welcome
Constructing irreducible polynomials recursively with a reverse composition method
We suggest a construction of the minimal polynomial of
over from the minimal polynomial for all positive integers whose prime factors divide . The
computations of our construction are carried out in . The key
observation leading to our construction is that for holds
where and
is a primitive -th root of unity in . The
construction allows to construct a large number of irreducible polynomials over
of the same degree. Since different applications require
different properties, this large number allows the selection of the candidates
with the desired properties
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