EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 1, 2021, 265-267 ISSN 1307-5543 – ejpam.com Published by New York Business Global Note on irreducible polynomials over finite field Amara Chandoul1,∗, Alanod M. Sibih2 1 Department of Mathematics Institut Supérieure d’Informatique et de Multimedia de Sfax, Sfax, Tunisia 2 Department of Mathematics, Jamoum University College, Umm Al-qura University, Saudi Arabia Abstract. In this note we extend an irreducibility criterion of polynomial over finite fields. We prove the irreducibility of the polynomial P (Y ) = Y n + λn−1Y n−1 + λn−2Y n−2 + · · ·+ λ1Y + λ0, such that λ0 6= 0, deg λn−2 = 2 deg λn−1 + l > deg λi, for all i 6= n− 2 and odd integer l. 2020 Mathematics Subject Classifications: 11R09, 11C08. Key Words and Phrases: Finite fields, Irreducible polynomials, Viéte theorem. 1. Introduction Irreducibility of polynomial functions seems like one of the major topics in introductory abstract algebra. It is not hard to see, using the fundamental theorem of algebra, that the only irreducible polynomials in C[x] are polynomials of degree 1. Then, it will be clear that the only irreducible polynomials in R[x] are polynomials of degree 1 and polynomials of the form ax2 + bx+ c with b2 − 4ac < 0 [4]. The situation over Q is much different from the situation over R or C. Over Q, there are many irreducible polynomials of every degree, and determining which polynomials are irreducible is difficult, compared to the real or complex case. Eisenstein’s criterion gives a sufficient condition for a polynomial with integer coeffi- cients to be irreducible over Q. This criterion is very nice when it works, but there are many irreducible polynomials to which it does not apply. Then, to decide irreducibility, one can try another approaches, like the so called ”brute force” [4]. In this paper, we will consider the setting over a finite field. Let, p be a prime and q a power of p. Let Fq be a finite field with q elements of characteristic p. It is known that there are no explicitly formula discribing irreduciblility of polynomials over Fq. Wherefore, we still need to provides methods to decide if a polynomial over Fq is irreducible. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3898 Email addresses: amarachandoul@yahoo.fr (A. Chandoul), amsibih@uqu.edu.sa (A. M. Sibih) http://www.ejpam.com 265 c© 2021 EJPAM All rights reserved. A. Chandoul, A. M. Sibih / Eur. J. Pure Appl. Math, 14 (1) (2021), 265-267 266 Irreducible polynomials over Fq, are used for many applications in mathematics. They are used to carry out the arithmetic in field extension of Fq [6]. Computations in such extensions occur in coding theory [1], complexity theory [7] and cryptography [5]. Random polynomial time algorithms exist for finding irreducible polynomials of any degree over Fq, [2, 7], and so as a practical matter the problem is solved. However, the deterministic complexity of the problem has yet to be established. In [2, 3], It was proved the following theorem, which gives an irreducibility criterion over Fq[X]. Theorem 1. Let P (Y ) = Y n+An−1Y n−1+An−2Y n−2+ · · ·+A0 with Ai ∈ Fq[X], A0 6= 0 and degAn−1 > degAi, for each i 6= n− 1. Then P (Y ) is irreducible over Fq[X]. In the following, we want to extend this result, in order to define a new family of irreducible polynomials over Fq[X]. 2. Main result We present the following result: Theorem 2. Let P (Y ) = Y n+λn−1Y n−1+λn−2Y n−2+· · ·+λ1Y +λ0 be a polynomial over Fq[X], such that λ0 6= 0, deg λn−2 > deg λi, for each i 6= n−2. If deg λn−2 = 2 deg λn−1+l, with l is odd, then P is irreducible. Proof. Using Viéte theorem, which establishes relations between the roots and the coefficients of a polynomial, one can easily see that if w = w1, w2, · · · , wn be the roots of P , then we have exactly 2 of its, with modulus strictly greater than 1. Suppose now, that P (Y ) = Q1(Y )Q2(Y ), where Q1 and Q2 are in Fq[X][Y ]. We can not suppose that the roots w1, w2 of P , with modulus strictly greater than 1 are roots of Q1, cause of the absolute value of the leading coefficient of the polynomial Q2 is superior or equal 1, which will be absurd because Q2 has only roots with modulus strictly less than 1. So, suppose that w1 is a root of Q1 and w2 is a root of Q2. Let P (Y ) = Q1(Y )Q2(Y ), with Q1(X) = Y s +As−1Y s−1 +As−2Y s−2 + · · · · · ·+A1Y +A0 and Q2(X) = Y m +Bm−1Y m−1 +Bm−2Y m−2 + · · ·+B1Y +B0. It is clear that degAs−1 > degAj , for all 1 ≤ j ≤ s, and degBm−1 > degBk, for all REFERENCES 267 1 ≤ k ≤ m. A simple calculation gives λn−1 = As−1 +Bm−1 and λn−2 = As−1Bm−1 +As−2 +Bm−2, which implies deg λn−1 = sup(degAs−1,degBm−1) and deg λn−2 = degAs−1 + degBm−1. One can easily see, that we have two cases: First case: If degAs−1 > degBm−1, then, we have deg λn−1 = degAs−1 and deg λn−2 < 2 deg λn−1, which is absurd because deg λn−2 = 2 deg λn−1 + l. Second case: If degAs−1 = degBm−1, then, we get deg λn−2 = 2 degAs−1, So deg λn−2 is even. But deg λn−2 = 2 deg λn−1 + l, then it is odd, which is the desired contradiction. Completing the proof. Remark The converse is not always true. Consider the Polynomial P (Y ) = Y 3 + (X2 + X)Y 2 + X3Y + 1 in F2[X][Y ]. P is an irreducible polynomial over F2[X] but deg(X3) 6= 2 deg(X2 +X) + l, for all l ∈ N∗. Acknowledgements All our thanks to the referees for careful reading of this manuscript and for theirs corrections and many important remarks. References [1] ER Berlekamp. Algebraic coding theory mcgraw-hill. New York, 8, 1968. [2] A Chandoul, M Jellali, and M Mkaouar. Irreducibility criterion over finite fields. Communications in Algebra, 39(9):3133–3137, 2011. [3] Amara Chandoul. Fractions continues multidimensionnelles: fractions continues mul- tidimensionelles, polynômes irréductibles et nombres de Pisot. Éditions universitaires européennes, 2012. [4] Lindsay N Childs. A concrete introduction to higher algebra. Springer, 2009. [5] Benny Chor and Ronald L Rivest. A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Transactions on Information Theory, 34(5):901–909, 1988. [6] Dirk Hachenberger and Dieter Jungnickel. Irreducible polynomials over finite fields. In Topics in Galois Fields, pages 197–239. Springer, 2020. [7] Michael O Rabin. Probabilistic algorithms in finite fields. SIAM Journal on computing, 9(2):273–280, 1980.