1. Introduction
Let
be a finite field, and let
n be a positive integer such that
. For any
, the
q-cyclotomic coset modulo
n containing
is defined by the subset
where
is the smallest positive integer such that
. Each element in
is called a representative of the coset
. The study of cyclotomic cosets plays a very important role in many computational problems. Since
q-cyclotomic cosets completely determine the irreducible factorization of
, and
over
, cyclotomic cosets are also needed in coding theory, such as the classification of constacyclic codes and counting problems involving constacyclic codes. These problems all require the parameters of cyclotomic cosets, for example, leaders, sizes, or representatives.
In [
1], the
q-cyclotomic cosets modulo
n contained in the subset
, such that
and
, were determined. Further, the authors enumerated the Euclidean self-dual codes. In [
2,
3], algorithms for calculating the leader of cyclotomic cosets were given based on the stream cipher
m-sequences and problems in statistical physics, respectively. Determining the representatives and the sizes of
q-cyclotomic cosets modulo
n under certain conditions can help in studying problems involving constacyclic codes. Some research work can be seen in [
4,
5,
6,
7,
8,
9,
10,
11].
This paper was inspired by [
12]. The purpose of this paper is to determine the number of irreducible factors of
over
for any
. We know that we can calculate the number of irreducible factors of
using the equation
where, for any
,
can be factored into
monic irreducible polynomials of the same degree
over
. In [
12], a necessary and sufficient condition for
was given, where
denotes the number of distinct irreducible factors of
over
. Suppose that
are
t factors of
n, and
denotes the order of
q modulo
. Then,
if and only if
is a solution to the Diophantine equation
As an application, [
12] obtained the sufficient and necessary conditions for
.
Through a careful study of the
q-cyclotomic cosets, we give more concrete factorization of
and
than the factorization formula given in [
13]. Then, based on the splitting or stability of cyclotomic cosets, we determine the number of irreducible factors of
and
over
for any
; however, the method differs from that in [
12]. As an application, we establish the necessary and sufficient conditions when
In addition, a more concrete factorization of
in these cases is given.
2. Preliminaries
Throughout this paper, we assume that n is a positive integer and that denotes a finite field with q elements, where is a power of a prime p and . is the multiplicative group of consisting of all non-zero elements of . We first review some basic results in number theory and finite fields, which we will use in the subsequent sections.
2.1. Basic Number Theory
Let
ℓ be a prime number. We denote the
ℓ-adic valuation of
n as follows:
Then we introduce the following lift-the-exponent lemmas in two cases.
Lemma 1 ([
2])
. Let ℓ be an odd prime number, and let m be an integer such that . Then, for any positive integer d. Lemma 2 ([
2])
. Let m be an odd integer, and d be a positive integer. Based on this, the following cases arise:- (1)
If , then - (2)
If and d is odd, then - (3)
If and d is even, then
2.2. Cyclotomic Cosets
If the decomposition of
n is given by
, where
are distinct prime numbers and
are positive integers,
m is an integer coprime to
n. We define
and
as follows:
If is a cyclic group, each generator of is called a primitive root modulo n. A well-known theorem about the existence of primitive roots is shown below.
Lemma 3 ([
14])
. Let n be a positive integer. Then, n possesses primitive roots if and only if n is of the form or , where p is an odd prime and α is a positive integer. We also review some familiar results below.
Lemma 4 ([
15])
. Let η be a primitive root modulo an odd prime ℓ. If is not congruent to 1 modulo , then η is a primitive root modulo for all . Lemma 5 ([
12])
. Let ℓ be an odd prime and η be a primitive root modulo , with . Then, η is a primitive root modulo for all . Lemma 6 ([
12])
. Let η be a primitive root modulo . Then η is a primitive root modulo , where , and d are defined as above. Recently, a criterion for the primitive root system and a proof of the existence of the primitive root system modulo
were given in [
16].
Lemma 7 ([
16])
. Let be distinct odd primes, and let be positive integers. An s-tuple of integers is said to be a primitive root system modulo if, for any , the following conditions hold:- (1)
is a primitive root modulo for all ;
- (2)
for any .
For any odd primes and positive integers , there exists a primitive root system modulo .
Further, the explicit representatives and sizes of q-cyclotomic cosets modulo an odd integer n are given. If for distinct odd primes different from p and positive integers , fix a system of primitive roots modulo n. For any integers such that , , the notations are defined as follows:
- (1)
;
- (2)
;
- (3)
.
Lemma 8 ([
16])
. Let be distinct odd prime numbers different from p, and let be positive integers. Then, all the distinct q-cyclotomic cosets modulo are given byfor , ; ; and , . By applying the lift-the-exponent lemma, a more concrete size of the q-cyclotomic cosets modulo n is given.
Lemma 9 ([
16])
. Denote by the set of all applicable -tuples that satisfy the inequality above. For any -tuple , the q-cyclotomic coset has a size ofwhere and , . Based on the theories above, [
16] considered the case where modulo
n is even. Let
, where
are different odd primes and
are positive integers. Since there is no primitive root of
, where
, Lemma 8 does not apply in this case.
Lemma 10 ([
16])
. Let n be an arbitrary positive integer, and let q be an odd prime power that is coprime to n. Let γ be an element in with the associated q-cyclotomic coset modulo n given by- (1)
If , that is, , then by viewing γ as an element in , the q-cyclotomic coset modulo containing γ is Moreover, the q-cyclotomic coset modulo containing is disjoint with , and the union is exactly the preimage of under the projection .
- (2)
If , that is, , then by viewing γ as an element in , the q-cyclotomic coset modulo containing γ is Moreover, the coset contains and is exactly the preimage of under the projection .
If fits the first condition in Lemma 10, we say that is splitting with respect to the extension . Otherwise, we say that is stable with respect to the extension . We define that an element is divisible by if any representative of in is divisible by . This definition is clearly well defined. Then, we give an explanation of q-cyclotomic cosets modulo n, where n is even.
For any
, where
is the greatest odd factor of
n, we denote the size of the
q-cyclotomic coset modulo
containing
by
, and it is easy to see that
. Since
we have that the
q-cyclotomic cosets modulo
containing
is
and
If
is odd, then we have
; otherwise, we have
. We assume that
(if
, the proof is similar). Since
we have that the
q-cyclotomic cosets modulo
containing
is
and
If
, then we can easily see that
. Conversely, if
, then we can easily see that
. So there must be one of
and
that is divisible by
. And so on, for
, if
, we can verify that there is exactly one among
and
that is divisible by
. Furthermore, there is a
q-cyclotomic coset that will keep splitting, while others will remain stable when
2.3. The Factorization of , , and
In this subsection, we review the formula in [
13] for the factorization of
,
, and
. Here,
n is a positive integer coprime to
q.
Lemma 11 ([
13])
. Let , and set if or ; otherwise, set . Furthermore, for all positive integers t, we define , and for every , where , denote a complete set of representatives of the q-cyclotomic cosets modulo , we set . Then, the factorization of into monic irreducible factors over iswhere, for all and all such that , the monic irreducible factor belonging to has degree and order . Lemma 12 ([
13])
. Let such that . Let , and set if or ; otherwise, set . Furthermore, we define . Then, the factorization of the cyclotomic polynomial into monic irreducible factors of degree over is Lemma 13 ([
13])
. Let and such that and , where and . Let , and set if or ; otherwise, set . For all positive integers t, we set and . If or , then ; otherwise, . For every , where denote a complete set of representatives of the q-cyclotomic cosets modulo , we set and . Then, there exists such that , and the factorization of over iswhere, for all applicable , the monic irreducible polynomial of degree and order is defined aswhere r is a positive integer satisfying if or otherwise. Furthermore, for all , we have if and only if for an integer It is not difficult to observe that, although [
13] provided the explicit factorization of
, and
, we cannot calculate the number of irreducible factors for any
such that
because the author did not provide the explicit representatives and sizes of the
q-cyclotomic cosets modulo
n in the formula.
3. Main Results
Let be a finite field, and let , n be a positive integer such that . Then, we give a different formula to calculate the number of irreducible factors of and over for any .
3.1. The Number of Irreducible Factors of
Let
, and set
,
.
Without loss of generality, we assume that the odd primes
are ranked in such a way that
are even, and
are odd.
Fix a system of primitive roots modulo , and let be the set of applicable -tuples, i.e., the tuples , such that and that and for .
For any , define to be the set of all s-tuples of integers satisfying and given . For any , where , let consisting of index i such that .
Now, we will calculate the number of irreducible factors of . We begin with the easier cases where, and = 1.
Proposition 1. - (1)
If , then can be factored into the irreducible factors over aswhere, for every and , the irreducible polynomial is given by - (2)
The number of irreducible factors of is
Proposition 2. - (1)
If , then can be factored into the irreducible factors over aswhere, for any and , the irreducible polynomials and are, respectively, given byand - (2)
The number of irreducible factors of is , where
We still need to consider the case where . Fix an applicable -tuple and an s-tuple . Let , .
Proposition 3. If , for any s-tuple , where , if and only ifwhere is the size of the q-cyclotomic coset modulo . Proof. Since
,
if and only if
is odd. According to Lemma 9, we know that
where
and
,
. Noting that
are even,
is odd if and only if
for any
. □
Proposition 4. For any , where , if , the number of irreducible factors of associated with is Proof. For any
, when
, there will be two
q-cyclotomic cosets modulo
d
generated by
, and there is one among
and
whose elements are all even, while the other has elements that are all odd. Suppose that
is odd, then
is even. Since
according to Lemma 10, we know it can split until
, i.e., when
, there will be
q-cyclotomic cosets modulo
d generated by
, and the length of them will be
. On the other hand, since
according to Lemma 10,
can also split until
, i.e., when
, there will be
q-cyclotomic cosets modulo
d generated by
, and the elements in these
q-cyclotomic cosets are all even. Noting that
is the subset of
such that
for
, we can easily see that when
, there will be
irreducible factors of
associated with
.
When
, the
q-cyclotomic cosets modulo
d generated by
are stable, and the
q-cyclotomic cosets modulo
d generated by
will split into
q-cyclotomic cosets. Now, we only need to consider the number of
q-cyclotomic cosets modulo
d generated by
. When
, there will be
stable
q-cyclotomic cosets and the other
q-cyclotomic cosets will split into
q-cyclotomic cosets modulo
d. Continuing this process, when
, there will be
q-cyclotomic cosets modulo
d. Furthermore, the number of irreducible factors of
associated with
is
If is even and is odd, the proof is similar. Now we conclude the proof of this proposition. □
Proposition 5. For any , where , if , the number of irreducible factors of associated with is Proof. For any
, when
, there will be two
q-cyclotomic cosets modulo
d
generated by
, and there is one among
and
whose elements are all even, while the other has elements that are all odd. Suppose that
is odd and
is even. Since
it will be stable when
, and the size of
is
. From the lift-the-exponent lemma,
and we can easily see that the
q-cyclotomic cosets
will split until
, i.e., when
, there will be
q-cyclotomic cosets modulo
d generated by
. On the other hand, since
for any
, there will be
q-cyclotomic cosets modulo
d generated by
. If all elements in these
q-cyclotomic cosets are even, we have that the number of the irreducible factors of
is
When
, the
q-cyclotomic cosets modulo
d generated by
are stable, and the
q-cyclotomic cosets modulo
d generated by
will split into
q-cyclotomic cosets modulo
d. When
, we only need to consider the
q-cyclotomic cosets modulo
d generated by
and when
, there will be
q-cyclotomic cosets that are stable, and the other
q-cyclotomic cosets will split into
q-cyclotomic cosets modulo
, and so on. When
, there will be
q-cyclotomic cosets modulo
d generated by
Furthermore, the number of irreducible factors of
associated with
is
If is even and is odd, the proof is similar. Now we conclude the proof of this proposition. □
Theorem 1. The notations are defined above. When , the number of irreducible factors of is When , the number of irreducible factors of is Proof. If
, for any
, where
,
. Then, by applying Lemma 8, we have that the number of
q-cyclotomic cosets modulo
d are
By applying Proposition 4, we can easily see that the irreducible factors of
when
are
When
, for any
, where
, if
is odd, then
; if
is even, then
. From Proposition 3 and similar to
, we have that the number of irreducible factors of
when
is
□
3.2. The Number of Irreducible Factors of
Let
, where
,
, and let
. Set
,
,
. If
or
, then
; otherwise,
.
m is a positive integer satisfying
if
or
.
Without loss of generality, we also assume that the odd primes
are ranked in such a way that
are even, and
are odd.
Fix a system of primitive roots modulo , Let be the set of applicable -tuples, i.e., the tuples , such that and and for .
For any , define as the set of all s-tuples of integers satisfying and when . For any , where , let consist of indices i such that ,
Now, we calculate the number of irreducible factors of . We begin with the easier cases where and = 1.
Proposition 6. - (1)
If , then can be factored into the irreducible factors over aswhere, for every , , , , the irreducible polynomial is given by - (2)
The number of irreducible factors of is
Proposition 7. - (1)
If , then can be factored into the irreducible factors over aswhere, for every , , , , the irreducible polynomial is given byand - (2)
The number of irreducible factors of is , where
We still need to consider the case where . Fix an applicable -tuple and an s-tuple . Let , . By combining Proposition 3, Proposition 4, Proposition 5, and Lemma 9, we can easily obtain the theorem below.
Theorem 2. The notations are defined above. When , the number of irreducible factors of is When , the number of irreducible factors of is