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

Sunil K. Chebolu, JáN Mináč

2011
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Mathematics Magazine
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369 Summary It is well known that the order statistics of a random sample from the uniform distribution on the interval [0, 1] have Beta distributions. In this paper we consider the order statistics of a random sample of n data points chosen from an arbitrary probability distribution on the interval [0, 1]. For integers k and with 1 ≤ k < ≤ n we find an attainable upper bound for the expected difference between the order statistics Y and Y k . This upper bound depends on the choice of k and but
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... does not depend on the distribution from which the data are obtained. We suggest a possible application of this result and we discuss some of its special cases. Why there are exactly 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 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, in the case when q is a prime. We 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.

doi:10.4169/math.mag.84.5.369
fatcat:pt6lztnar5d7nelrbbqx4nlvje