On the Irreducibility of Polynomials Associated with the Complete Residue Systems in any Imaginary Quadratic Fields
For a Gaussian prime and a nonzero Gaussian integer with and , it was proved that if where , , belong to a complete residue system modulo , and the digits and satisfy certain restrictions, then the polynomial is irreducible in . For any quadratic field , it is well known that there are explicit representations for a complete residue system in , but those of the case are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.
According to Pólya and Szegö , the classical result of A. Cohn states that if a prime is expressed in the decimal representation asthen the polynomial is irreducible in . For example, since 4831 is prime and , it follows that is irreducible in . This result was subsequently generalized to any base by Brillhart et al. . In 2002, Murty gave a proof of this fact  that was conceptually simpler than the one in . Moreover, Brillhart et al.  generalized Cohn’s result in another direction as follows: ifwhere for all , and is prime, then is irreducible. In 1988, Filaseta improved this fact, that is, if coefficients of the polynomial in (2) satisfy for and is prime, then is irreducible .
In another direction, let , with a unique squarefree integer unequal to 1, be a quadratic field and be its ring of integers. We have seen in  that is an integral domain. The quadratic field is said to be real if and imaginary if . Clearly, is real if and imaginary if . It is well known thatwhere
We note that the ring of Gaussian integers and the Eisenstein domain are rings of integers of and , respectively.
We emphasize that a prime element in is an irreducible element, and the converse holds if is a unique factorization domain. Moreover, is the quotient field of , and the units in are the units in . We say that a nonzero polynomial is irreducible in if is not a unit, and if with in , then or is a unit in . Polynomials that are not irreducible are called reducible. Furthermore, if is a unique factorization domain and is irreducible in , then is irreducible over by using Gauss’s lemma.
In 2017, Singthongla et al. established the result of A. Cohn in , where is an imaginary quadratic field such that is a Euclidean domain, namely, , and . The results are as follows:(i)For , each element in has a base -expansion whose digits are bounded by certain constants.(ii)For a prime element in , if is its base -expansion in and the digits and satisfy some natural restrictions, then the polynomial is irreducible in .(iii)For Gaussian integers, a similar base -expansion but with digits belonging to a complete residue system modulo is also valid.(iv)The irreducibility result similar to that in (ii) continues to hold for Gaussian integers with base expansion as described in (iii). At first thought, we are interested in establishing a generalization of the result in (iv) for the ring of integers of any general quadratic field. However, to prove this, we must use the property that for all which is valid only in the imaginary quadratic field  but not in the real quadratic field. Thus, the goal of this work is to establish a generalization of the result in (iv) for the ring of integers of any imaginary quadratic field.
Let be any general quadratic field. For with , we say that divides , denoted by , if and only if there exists such that . For with , we say that is congruent to modulo , and we write if , a principal ideal generated by or, equivalently, if .
For , we denote the norm of by
We note from  that if is such that , where is a rational prime, then is an irreducible element. By a complete residue system modulo in , abbreviated by [9, 10], we mean a set of elements in such that(i)for each , there is such that and(ii) for all , with
In , Tadee et al. derived three explicit representations for a complete residue system in any general quadratic field. We are interested in the first one which is as follows: for and , we have the following:(i)Case : the set is a .(ii)Case :(1)If is even, let Then, is a .(2)If is odd, let
Then, is a .
We observe that the complete residue system for the case in  is complicated and inapplicable for our work. In this paper, we first establish a complete residue system for the case which is similar to that in (7). Then, we determine the so-called base -representation in and generalize the result in (iv) for the ring of integers of any imaginary quadratic field by using such representation.
2. Complete Residue System for the Case
In this section, we first establish a complete residue system for any quadratic field , with as in the following theorem.
Theorem 1. Let be a quadratic field, with a squarefree integer unequal to 1 such that . If , with , then the setis a .
Proof. Since , there exist such that . We first note thatLet . Then, by the division algorithm, there exist such that , where . Then,It follows from (11) and (12) thatAgain, by the division algorithm, there exist such thatwhere . It follows from (13) and (14) thatwhere . We consequently get , and so, , where .
Next, we show that for all , with . To see this, let be such that . We now show that . Since and in , we have , and so, . It follows that because . Consequently, , and so, .
If and , we obtain that , , and . Thus, there exist such thatHence, , and so, . It follows that because . Similarly, if and , then , , and . Since , we have because .
For the case and , since , there exist such that , , and . From in , there exist such thatIt follows that , that is,Since , we have and . Thus, there exist such thatIt follows from (18) and (19) that , and so, . Now, we have thatso . Since , it follows that .
From all cases, we deduce that , and so, , as desired.
Remark 1. Keeping the notation of Theorem 1 with , we have that
Proof. To show that , we must show thatIt is easy to see that inequality (22) holds if or . We now assume that and . Note that , , and . Since and , we haveand (22) is proved.
For a quadratic field , with and , we recall thatis a , where . It is clear that whenever . It follows thatFrom both cases, we observe that if , then , while if , then .
We end this section by determining the so-called base -representation.
Definition 1. Let be an imaginary quadratic field. Let and be a . We say that has a base -representation iffor some , , and . If , then (26) is called a base -representation of .
Example 1. Let , , and . Then, and , and so,is a . Since , there exists a unique such that . We see that andSince , there exists a unique such that . We see that andIt follows from (28) and (29) thatSince , there exists a unique such that . We see that andIt follows from (30) and (31) thatSince , there exists a unique such that . We see that andIt follows from (32) and (33) thatContinuing the process, we conclude that can be written as a base -representation in infinite ways as (28), (30), and (32), andfor all .
Since , we note that (28) and (30) are two base -representations of .
Example 2. Let , , and . Then, and , and so,is a . By using the process as in the previous example, we obtain thatSince , we get that the process stops, and (37)–(40) are four base -representations (base -representations) of .
Example 3. Let , , and . Then, and , and so,is a . By using the process as in the previous examples, we conclude that can be written as a base -representation in infinite ways asSince , we note that (42)–(44) are three base -representations of .
3. Irreducibility Criteria for Polynomials over Imaginary Quadratic Fields
We first recall the result in (iv) as in the following theorem .
Theorem 2. Let or be such that and . For a Gaussian prime , ifis its base -representation with and satisfying the condition , then is irreducible in .
Lemma 1. Let be an imaginary quadratic field. If , then . We note for an imaginary quadratic field that for all , the group of units in .
Lemma 2. Let be such that and for some positive real number . If satisfies(i) and(ii), then any complex zero of satisfies either or .
3.1. Irreducibility Criterion for
To obtain an irreducibility criterion for the case , we begin with the following lemmas.
Lemma 3. Let be an imaginary quadratic field, with . Let be such that , and letwhere . Then, .
Proof. If , then and . It follows thatNow, we assume that , so there are two possible cases.
Case 1. . We have and . Since , we obtain , and so,It follows from (49) thatThus,which implies thatLet .
We will show that . If , then . Since , we obtain that