Abstract
The problem of finding the number of irreducible monic polynomials of degree over is considered in this paper. By considering the fact that an irreducible polynomial of degree over has a root in a subfield of if and only if , we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of . We also use the lattice of subfields of to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of .
1. Introduction
In this paper we consider the problem of finding the number, , of monic irreducible polynomials of degree over the field , where is a positive integer and is the power of a prime number. This problem has been discussed by several authors including C. F. Gauss who gave the following beautiful formula: where runs over the set of all positive divisors of including 1 and and is the Möbius function; see [1]. Recently, it has been shown, see [2], that this number can be found by using only basic facts about finite fields and the Principle of Inclusion-Exclusion. This work seeks to emphasize the simplicity of the method given in [2] by using a lattice of subfields. This is done by first of all proving Gauss’s formula using the Principle of Inclusion-Exclusion as was done in [2]. However, we use only one basic fact about where (in which subfields) the roots of irreducible polynomials of degree over can lie. We then show how a lattice of subfields of the field, , can be used to obtain . We are particularly interested in the number of roots of irreducible polynomials of degree over because the problem of counting irreducible Goppa codes of length and of degree depends on this number.
2. Preliminaries
2.1. The Number of Irreducible Polynomials
Our approach to counting the number of irreducible polynomials of degree over is to count the number of all roots of such polynomials. To this end, we make the following definitions.
Definition 1. One defines the set to be the set of all elements in of degree over .
Definition 2. One defines the set to be the set of all irreducible monic polynomials of degree over .
The following theorem is well known.
Theorem 3. is given by formula (1).
For the sake of clarity we state the relationship between and which immediately leads to the “Gaussian like” count of the number of elements in . We put this in the following corollary.
Corollary 4. is the union of all the roots of the polynomials in and
2.2. Where Elements of Lie
We next identify the subfields of where the elements of lie. To achieve this we first note that an irreducible polynomial over may, in some specific cases, be seen as irreducible over an extension field of . To be more specific, we state this in the following theorem.
Theorem 5. An irreducible polynomial over of degree remains irreducible over if and only if ; see [3].
Now, in order to apply Theorem 5 to general cases, one makes the following decompositions of and .
Definition 6. One defines to be the largest divisor of that is relatively prime to and set and defines to be the largest divisor of that is relatively prime to and set . Thus , where .
With this notation, the following lemma is a direct result of Theorem 5.
Lemma 7. consists of the elements of each of which is a root of an irreducible polynomial of degree over , where is a divisor of . In particular, the elements of are precisely those elements that lie in a subfield of of the form , for some , but not in any subfield of the form , where is not divisible by . See [4].
It is useful, for our purposes, to think of the subfields identified in Lemma 7 in the following way.
Corollary 8. The subfields of the form defined in Lemma 7 are the maximal subfields of such that or the subfields contained in such maximal subfields.
2.3. The Principle of Inclusion-Exclusion
Since we will be making extensive use of the “Principle of Inclusion-Exclusion” we state this well known principle in the following theorem; see [5].
Theorem 9. Let be finite sets. Then
2.4. Goppa Codes
This paper is motivated by the unsolved problem of finding an irreducible polynomial which defines a “good” Goppa code of degree and length or equivalently finding an element of degree which defines such a code. So it is appropriate for us to define a Goppa code. The following definition is the classical definition found in much of the literature on coding theory.
Definition 10. Let be irreducible of degree and let . Then the irreducible Goppa code is defined as the set of all vectors with components in which satisfy the condition
The polynomial is called the Goppa polynomial. Since is irreducible and of degree over , does not have any root in and the code is called an irreducible Goppa code of degree . In this paper is always irreducible of degree over .
2.4.1. Irreducible Goppa Codes Defined by a Field Element
The following characterization of an irreducible Goppa code is particularly useful for our purposes. It can be shown, see [6], that if is any root of the Goppa polynomial then is completely described by any root of and a parity check matrix is given by We denote this code by . Since is completely described by the root of the number gives an upper bound on the number of irreducible Goppa codes. Furthermore, knowing the various locations (subfields) of the elements of will facilitate research into finding the element which will give the best Goppa code.
3. Proof of Gauss’s Formula
We now give a new proof of Gauss’s Formula when applied to find the cardinality of the set which has special significance in the application to Goppa codes. Putting in our proof will give the result proved in [2]. There are many similarities between approach given in [2] and our method. However, the crux of our argument lies on Corollary 8 which in turn is based on Theorem 5. We believe that this slightly different approach brings a little more clarity to the situation.
Proof. Let be the prime factorisation of . The maximal subfields of , as in Corollary 8, are of the form Thus, by Corollary 8, , where the complement is taken in . As in [2], we note that , , and so forth. Using the Principle of Inclusion-Exclusion, it follows that Finally, note that .
4. Using a Lattice of Subfields of
We have noted that in order to construct a lattice of subfields of together with Corollary 8 offers a good insight into . We will put the lattice of subfields of into their usual hierarchies, where level one is taken by , and level two contains subfields of of the form , where is the prime factorization of . Level three comprises of maximal subfields of the fields in level two and so on. As this is done, all intersections between subfields are marked as this is used in getting .
Observe that the elements of (those not lying in ) described in Corollary 8 are those which lie in subfields of type , where is a prime divisor of . We can see the formula for taking shape as we have the full splitting field of all the irreducible polynomials of degree over , the maximal subfields , and the subfields of .
Formula (7) shows the levels mentioned above. The subfields that need to be considered in order to find can be read off from a lattice of subfields of in accordance with Corollary 8, that is, subfields of the form . We illustrate this method with an example.
Example 11. Let us take , , and . Then, , , , and . Hence the subfields in level two which do not contain any elements of (subfields of the form ) are and . While the only proper subfield which does contain elements of (subfield of the form ) is , putting . So in constructing the set it is necessary to exclude the two subfields and . In level three, we will consider the intersection as this has been excluded twice. Thus, the number of elements of degree 6 over is . The lattice shown in Figure 1 illustrates this example.
Example 12. Let us take , , and . In this case and . There are no proper subgroups of the form but rather all maximal subgroups are of the form and so by Corollary 8 all these maximal subgroups are excluded when constructing the set . By the Principle of Inclusion-Exclusion, it follows that . See Figure 2.
5. Applications to Goppa Codes
We know that an irreducible Goppa code, , is defined by a root of the Goppa polynomial which is of degree over . Now to find such an one can search in any of the subfields of the form . This makes the search easier. We can use a lattice of subfields of not only to facilitate this search but also to calculate the number of such elements.
Example 13. Let us look again at the example above with , , and . contains elements of . The number of elements in can be easily calculated from the lattice of subfields. It is . See Figure 3.
6. Conclusion
In this paper we have shown how a lattice of subfields can be used as an alternative to Gauss’s formula for finding the number of monic irreducible polynomials of degree over . The lattice of subfields approach helps to clear the mystery surrounding the rather complicated looking Gaussian formula which involves the Möbius function. We have also shown how this method can be used to obtain a Goppa code , where lies in a lower field. Using the Principle of Inclusion-Exclusion with the lattice of subfields it is easy to calculate the number of such elements in any subfield. The lattice of subfields approach simplifies the task of finding Goppa codes and sheds light on the processes involved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.