Counting Irreducible Polynomials over Finite Fields Using the Inclusion-Exclusion Principle

Abstract

C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Using just very basic knowledge of finite fields and the inclusion-exclusion formula, we show how one can see the shape of this formula and its proof almost instantly.Why there are exactly 1 30 (2 30 − 2 15 − 2 10 − 2 6 + 2 5 + 2 3 + 2 2 − 2)irreducible monic polynomials of degree 30 over the field of two elements? In this note we will show how one can see the answer instantly using just very basic knowledge of finite fields and the well-known inclusion-exclusion principle.To set the stage, let F q denote the finite field of q elements. Then in general, the number of monic irreducible polynomials of degree n over the finite field F q is given by Gauss's formula 1 n d|n µ(n/d)q d , where d runs over the set of all positive divisors of n including 1 and n, and µ(r) is the Möbius function. (Recall that µ(1) = 1 and µ(r) evaluated at a product of distinct primes is 1 or -1 according to whether the number of factors is even or odd. For all other natural numbers µ(r) = 0.) This beautiful formula is well-known and was discovered by Gauss [2, p. 602-629] in the case when q is a prime. We will present a proof of this formula that uses only elementary facts about finite fields and the inclusion-exclusion principle. Our approach offers the reader a new insight into this formula because our proof gives a precise field theoretic meaning to each summand in the above formula. The classical proof [3, p. 84] which uses the Möbius' inversion formula does not offer this insight. Therefore we hope that students and users of finite fields may find our approach helpful. It is surprising that our simple argument is not available in textbooks, although it must be known to some specialists.

Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Latt

“…We now give a new proof of Gauss's Formula when applied to find the cardinality of the set S which has special significance in the application to Goppa codes. Putting = 1 in our proof will give the result proved in [2]. There are many similarities between approach given in [2] and our method.…”

“…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.…”

“…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].…”

“…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 F can lie.…”

Mathematical Method to Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)

“…Many of BP(x), which has an elemental polynomial as a factor under GF(p q ), are termed as reducible. Those of the BP(x) that have no factors are termed as irreducible polynomials IP(x) [3] [4] and is expressed as, IP(x) = a q x q + a q-1 x q-1 + ---+ a 1 x + a 0 , where a q ≠ 0.…”