The Number of Irreducible Polynomials of Even Degree over F 2 with the First Four Coefficients Given

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1 The Number of Irreducible Polynomials of Even Degree over F 2 with the First Four Coefficients Given B. Omidi Koma School of Mathematics and Statistics Carleton University bomidi@math.carleton.ca July 22, 2010 B. Omidi Koma () Irreducible Polynomials July 22, / 34

2 1 Introduction Definitions and Background Previous Results 2 Generalizing Möbius Inversion Formula 3 Computing F(n,t 1,t 2,t 3,t 4 ) 4 The Formula For N(n,t 1,t 2,t 3,t 4 ) 5 Cosets of F 2 l in F 2 n 6 An Approximation of N(n,t 1,t 2,t 3,t 4 ) 7 Some Outputs of Our Approximation 8 Future Directions 9 References B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and2 Constan / 34

3 Definitions and Background For β F 2 n the k th trace is denoted by T k (β), and defined as T k (β) = β i 1 β i2 β i k. 0 i 1 <i 2 < <i k n 1 F(n,t 1,,t r ) : the number of β F 2 n where T i (β) = t i, 1 i r. Let f (x) = x n + a n 1 x n a 1 x + a 0 be a polynomial of degree n over F 2. a n i is the i th trace of f, written as T i (f ) = a n i, for 1 i n. P(n,t 1,,t r ) : the set of irreducible polynomials of degree n over F 2, where a n i = t i F 2. Let N(n,t 1,,t r ) = P(n,t 1,,t r ). B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and3 Constan / 34

4 Previous Results Let N(n,q) be the number of monic irreducible polynomials of degree n over F q [x]. Then q n = dn(n,q). By Möbius inversion formula we have N(n,q) = 1 n ( n ) µ q d = 1 d n µ(d)q n d. The number of irreducible polynomials with the first coefficient prescribed is given by Carlitz (1952). B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and4 Constan / 34

5 Let n be an even integer, and a b (mod 4) be shortened to a b. Then Cattell, Miers, Ruskey, Serra, and Sawada, (2003) proved that nn(n,1,0) = µ(d)f(n/d,1,0) + nn(n,1,1) = µ(d)f(n/d,1,1) + µ(d)f(n/d,1,1), µ(d)f(n/d,1,0), nn(n,0,0) = nn(n,0,1) = d odd d odd µ(d)f(n/d,0,0) µ(d)f(n/d,0,1),n/deven d odd,n/deven d odd µ(d)2 n 2d 1, µ(d)2 n 2d 1. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and5 Constan / 34

6 Also F(n,t 1,t 2 ) = 2 n 2 + G(n,t 1,t 2 ), and for n = 2m we have Table: Values of G(n, t 1, t 2 ) m (mod 4) (0, 0) (0, 1) (1,0) (1,1) 0 2 m 1 2 m m 1 2 m m 1 2 m m 1 2 m 1 The number of irreducible polynomials of even degree n over F 2 with given first three coefficients is considered by Yucas and Mullen (2004). For odd degree n Fitzgerald and Yucas (2003) consider the same problem. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and6 Constan / 34

7 Generalizing Möbius Inversion Formula For our study we proved the following generalization of Möbius Inversion. Theorem Suppose a b (mod 8) be written as a b. Let A(n), B(n), C(n), D(n), α(n), β(n), γ(n), and δ(n) be functions defined on N. Then α + X β + X γ + X δ, d d d d d 5 d 7 if and only if A(n) = X B(n) = X C(n) = X D(n) = X β + X d γ + X d δ + X γ d α + X d d 5 δ + X d d 5 δ + X d α + X β d d 5 d 7 d + X d 7 + X α d d 7 γ, d β, d d, B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and7 Constan / 34

8 Theorem (Countinued) α(n) = X β(n) = X γ(n) = X δ(n) = X µ(d)a + X µ(d)b + X µ(d)c + X µ(d)d, d d d d d 5 d 7 µ(d)b + X d µ(d)c d + X µ(d)d + X µ(d)c d µ(d)a + X d d 5 µ(d)d + X d d 5 + X µ(d)b d d 5 µ(d)d + X d d 7 µ(d)a + X d d 7 µ(d)c d, µ(d)b, d + X µ(d)a d d 7 d. In general, we proved that when modulo is 2 j, where j > 3, we have 2 j 1 functions in terms of some other 2 j 1 functions. For our case, j = 3. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and8 Constan / 34

9 Computing F(n, t 1, t 2, t 3, t 4 ) Lemma Let β F 2 n and f F 2 [x] of degree n/d be the minimal polynomial of β. Then the i th trace of β is the coefficient of x n i in f d, or T i (β) = T i (f d ), where 1 i n. Lemma Let d 1 be an integer, and f F 2 [x]. Then (i) T 1 (f d ) = dt 1 (f ); ( ) d (ii) T 2 (f d ) = T 1 (f ) + dt 2 (f ); 2 ( ) d (iii) T 3 (f d ) = T 1 (f ) + dt 3 (f ); 3 ( ) ( ) d d (iv) T 4 (f d ) = T 1 (f ) + T 2 (f ) + dt 4 (f ). 4 2 B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with July Given 22, 2010 Trace and9 Constan / 34

10 Lemma Let d be a given integer such that d 1. For 0 j 7 we shorten d j (mod 8) as d j. If f F 2 [x] is a given polynomial, then for 0 j 7 the coefficients T i (f d ) where i = 1,2,3,4 are given in the following table. Table: Different coefficients of f d d i T 1(f d ) T 2(f d ) T 3(f d ) T 4(f d ) d d 1 T 1(f ) T 2(f ) T 3(f ) T 4(f ) d 2 0 T 1(f ) 0 T 2(f ) d 3 T 1(f ) T 1(f ) + T 2(f ) T 1(f ) + T 3(f ) T 2(f ) + T 4(f ) d T 1(f ) d 5 T 1(f ) T 2(f ) T 3(f ) T 1(f ) + T 4(f ) d 6 0 T 1(f ) 0 T 1(f ) + T 2(f ) d 7 T 1(f ) T 1(f ) + T 2(f ) T 1(f ) + T 3(f ) T 1(f ) + T 2(f ) + T 4(f ) B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and10 Constan / 34

11 Theorem Let n be an even positive integer and t i F 2, for 1 i 4. Also let a b (mod 8) be shortened as a b. Then the number of elements β F 2 n with prescribed first four traces T i (β) = t i can be given by F(n,t 1,t 2,t 3,t 4 ) = 7 i=0 d i n d. S i, where the sets S 0,...,S 7 are defined as: S 0 = {f P(n/d) : t i = 0, i = 1,2,3,4}, S 1 = {f P(n/d) : T i (f ) = t i, i = 1,2,3,4}, S 2 = {f P(n/d) : t 1 = t 3 = 0, T 1 (f ) = t 2, T 2 (f ) = t 4 }, B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and11 Constan / 34

12 Theorem (Continued) S 3 = {f P(n/d) : T 1 (f ) = t 1, T 1 (f ) + T i (f ) = t i, i = 2,3, T 2 (f ) + T 4 (f ) = t 4 }, S 4 = {f P(n/d) : t i = 0, i = 1,2,3, T 1 (f ) = t 4 }, S 5 = {f P(n/d) : T i (f ) = t i, i = 1,2,3, T 1 (f ) + T 4 (f ) = t 4 }, S 6 = {f P(n/d) : t 1 = t 3 = 0, T 1 (f ) = t 2, T 1 (f ) + T 2 (f ) = t 4 }, S 7 = {f P(n/d) : T 1 (f ) = t 1, T 1 (f ) + T i (f ) = t i, i = 2,3, T 1 (f ) + T 2 (f ) + T 4 (f ) = t 4 }. We use the last theorem and our generalization of Möbius Inversion formula to find the number N(n,t 1,t 2,t 3,t 4 ). B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and12 Constan / 34

13 The Formula For N(n, t 1, t 2, t 3, t 4 ) Theorem For even degree n, different values of N(n,t 1,t 2,t 3,t 4 ) are given as (i) nn(n,1, 1,1, 0) = X µ(d)f(n/d,1, 1,1, 0) + X + X µ(d)f(n/d,1, 1,1, 1) + X d 5 d 7 nn(n,1, 0,0, 1) = X µ(d)f(n/d,1, 0,0, 1) + X + X µ(d)f(n/d,1, 0,0, 0) + X d 5 d 7 nn(n,1, 1,1, 1) = X µ(d)f(n/d,1, 1,1, 1) + X + X µ(d)f(n/d,1, 1,1, 0) + X d 5 d 7 µ(d)f(n/d,1,0, 0,0) µ(d)f(n/d,1,0, 0,1), µ(d)f(n/d,1,1, 1,0) µ(d)f(n/d,1,1, 1,1), µ(d)f(n/d,1,0, 0,1) µ(d)f(n/d,1,0, 0,0), B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and13 Constan / 34

14 Theorem (Continued) nn(n,1, 0, 0,0) = X µ(d)f(n/d,1,0, 0,0) + X (ii) nn(n,0, 0, 1,0) = X d odd (iii) nn(n,0, 0, 1,1) = X d odd + X µ(d)f(n/d,1,0, 0,1) + X d 5 d 7 µ(d)f (n/d,0,0, 1,0), µ(d)f (n/d,0,0, 1,1), (iv) nn(n,1, 1, 0,0) = X µ(d)f(n/d,1,1, 0,0) + X + X µ(d)f(n/d,1,1, 0,1) + X d 5 d 7 µ(d)f(n/d,1, 1,1, 1) µ(d)f(n/d,1, 1,1, 0), µ(d)f(n/d,1, 0,1, 0) µ(d)f(n/d,1, 0,1, 1), B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and14 Constan / 34

15 Theorem (Continued) nn(n,1, 0,1, 1) = X µ(d)f(n/d,1, 0,1, 1) + X + X µ(d)f(n/d,1, 0,1, 0) + X d 5 d 7 nn(n,1, 1,0, 1) = X µ(d)f(n/d,1, 1,0, 1) + X + X µ(d)f(n/d,1, 1,0, 0) + X d 5 d 7 nn(n,1, 0,1, 0) = X µ(d)f(n/d,1, 0,1, 0) + X + X µ(d)f(n/d,1, 0,1, 1) + X d 5 d 7 µ(d)f(n/d,1,1, 0,0) µ(d)f(n/d,1,1, 0,1), µ(d)f(n/d,1,0, 1,1) µ(d)f(n/d,1,0, 1,0), µ(d)f(n/d,1,1, 0,1) µ(d)f(n/d,1,1, 0,0). B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and15 Constan / 34

16 Theorem (Continued) In the following cases, a b represents a b (mod 4). (v) nn(n,0,1, 1,1) = X µ(d)f(n/d,0,1, 1,1) + X nn(n,0,1, 1,0) = X (vi) nn(n,0,0, 0,0) = X d odd (vii) nn(n,0,0, 0,1) = X d odd µ(d)f(n/d,0,1, 1,0) + X µ(d)f(n/d,0, 0,0, 0) X µ(d)f(n/d,0, 0,0, 1) µ(d)f(n/d,0, 1,1, 0), µ(d)f(n/d,0, 1,1, 1),, d n even d odd X, n d even d odd µ(d)f(n/2d, 0, 0), µ(d)f(n/2d, 0, 0), B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and16 Constan / 34

17 Theorem (Continued) and finally, (viii) nn(n,0, 1,0, 0) = X + X nn(n,0, 1,0, 1) = X + X µ(d)f(n/d,0,1, 0,0) µ(d)f(n/d,0,1, 0,1) µ(d)f(n/d,0,1, 0,1) µ(d)f(n/d,0,1, 0,0) X, d n even X, d n even X, d n even X, n d even µ(d)f(n/2d, 1,0) µ(d)f(n/2d, 1, 1), µ(d)f(n/2d, 1,1) µ(d)f(n/2d, 1, 0). B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and17 Constan / 34

18 Theorem (Continued) and finally, (viii) nn(n,0, 1,0, 0) = X + X nn(n,0, 1,0, 1) = X + X µ(d)f(n/d,0,1, 0,0) µ(d)f(n/d,0,1, 0,1) µ(d)f(n/d,0,1, 0,1) µ(d)f(n/d,0,1, 0,0) X, d n even X, d n even X, d n even X, n d even µ(d)f(n/2d, 1,0) µ(d)f(n/2d, 1, 1), µ(d)f(n/2d, 1,1) µ(d)f(n/2d, 1, 0). B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and18 Constan / 34

19 Cosets of F 2 l in F 2 n Let n = 4l. For a given α F 2 l the linear functional L α : F 2 l F 2 is defined as L α (β) = T 2 l(αβ), for all β F 2 l. We define W 0 = Ker(T 2 l) = {β F 2 l T 2 l(β) = 0}, and W i (γ) = {β F 2 l T i (β + γ) = T i (β) + T i (γ)}, for i = 1,...,4. For γ F 2 n there exist three possibilities for W i (γ), where i = 2,3,4. Either W i (γ) = F 2 l, or W i (γ) = W 0, or W i (γ) is a hyperplane different that W 0. In total there exist 27 different cases. We proved that some of these cases are possible. We divide them into three groups, based on W 2. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and19 Constan / 34

20 Table: Group one, W 2 (γ) = F 2 l W 3 W 4 F 2 l W 0 W 3 W 0, F 2 l F 2 l Cat A N/A N/A W 0 Cat A 1 N/A N/A W 4 W 0, F 2 l Cat A 2 N/A N/A Table: Group two, W 2 (γ) = W 0 W 3 W 4 F 2 l W 0 W 3 W 0, F 2 l F 2 l N/A N/A N/A W 0 Cat B 1 Cat C 1 N/A W 4 W 0, F 2 l Cat B 2 Cat C 2 N/A B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and20 Constan / 34

21 Table: Group three, W 2 (γ) W 0, F 2 l W 3 W 4 F 2 l W 0 W 3 W 0, F 2 l F 2 l N/A N/A N/A W 0 Cat F 1 Cat D 1 Cat E 1 W 4 W 0, F 2 l Cat F 2 Cat D 2 Cat E 2 For each category we have some conditions on γ F 2 n. For example, a coset γ + F 2 l is in D 1 iff ˆγ = γ + γ 2l + γ 22l + γ 23l 1 and either (i) T 2 l(ˆγ) = 0 and ˆγ 4 = ˆγ + 1, or (ii) T 2 l(ˆγ) = 1 and ˆγ 4 = ˆγ 2 + ˆγ + 1. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and21 Constan / 34

22 For γ F 2 n the coset γ + F 2 l is in one of the 13 categories. Then we can compute different traces of an element γ + β from the coset γ + F 2 l, where β F 2 l. For example if the coset γ + F 2 l is in category D 1, the traces of an element γ + β are given in the following table. Table: Traces in Category D 1 T i(β + γ) T 1(β + γ) β F 2 l T 2(β + γ) T 3(β + γ) T 4(β + γ) β W 2 W 0 T 1(γ) T 2(γ) T 3(γ) T 4(γ) β W 2 \ (W 2 W 0) T 1(γ) T 2(γ) T 3(γ) + 1 T 4(γ) β W 0 \ (W 2 W 0) T 1(γ) T 2(γ) + 1 T 3(γ) T 4(γ) β F 2 l \ (W 2 W 0) T 1(γ) T 2(γ) + 1 T 3(γ) + 1 T 4(γ) B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and22 Constan / 34

23 In different categories T i (γ + β), 1 i 4, is in terms of T i (γ), which is either zero or one. To complete the study, we need to know in each category for how many of the elements γ we have T i (γ) = 0, and T i (γ) = 1. We expect that T i (γ) depends on m, or l, where n = 2m = 4l. Once the conditions are obtained one is able to compute the the exact value of N(n,t 1,t 2,t 3,t 4 ), where n = 4l. Similar results to these studies are then needed for the case n = 4l + 2. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and23 Constan / 34

24 An Approximation of N(n, t 1, t 2, t 3, t 4 ) We have 16 cases for (t 1,t 2,t 3,t 4 ). In our main theorem these 16 cases are divided to 8 groups, and each group has some of the cases of (t 1,t 2,t 3,t 4 ) that are connected to each other. Then for different groups the formulas for the number N(n,t 1,t 2,t 3,t 4 ) are given in terms of F(n/d,t 1,t 2,t 3,t 4 ) s where d n, and (t 1,t 2,t 3,t 4 ) is from the same group as (t 1,t 2,t 3,t 4 ). Assume that F(n/d,t 1,t 2,t 3,t 4) = { 2 n d 4 if n d 4, 0 otherwise. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and24 Constan / 34

25 We let n = 2 k 0 k1 p 1 p k s s, where k 0,...,k s 1, and p 1,...,p s are odd prime divisors of n. Assume that S 1 = {p 1,...,p s }, and for 2 q s let S q = {d : d n,d is the product of exactly q distinct odd primes p i S 1 } For the first 12 cases of (t 1,t 2,t 3,t 4 ) which are in groups (i) to (v), the number N(n,t 1,t 2,t 3,t 4 ) can be given as N(n,t 1,t 2,t 3,t 4 ) = 1 2 n 4 + n s [n/d 4]( 1) q 2 n/d 4. d S q q=1 B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and25 Constan / 34

26 For (t 1,t 2,t 3,t 4 ) from groups (vi) we have (t 1,t 2,t 3,t 4 ) = (0,0,0,0) and N(n,0,0,0,0) = 1 s 2 n 4 + [n/d 4]( 1) q 2 n/d 4 n q=1 d S q 1 s (F(n/d,0,0) + ( 1) q F(n/2d,0,0). n q=1 d S q In group (vii) we have (t 1,t 2,t 3,t 4 ) = (0,0,0,1) and N(n,0,0,0,1) = 1 s 2 n 4 + [n/d 4]( 1) q 2 n/d 4 n q=1 d S q 1 s (F(n/d,0,1) + ( 1) q F(n/2d,0,1). n q=1 d S q B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and26 Constan / 34

27 Finally in group (viii), if (t 1,t 2,t 3,t 4 ) = (0,1,0,0) then the estimate for N(n,t 1,t 2,t 3,t 4 ) can be given as 1 s 2 n 4 + ( 1) q [n/d 4]2 n/d 4 F(n/2,1,0) n q=1 d S q 1 s ( 1) q ([d 1]F(n/2d,1,0) + [d 3]F(n/2d,1,1)), n q=1 d S q and if (t 1,t 2,t 3,t 4 ) = (0,1,0,1) the estimate for N(n,t 1,t 2,t 3,t 4 ) is 1 s 2 n 4 + ( 1) q [n/d 4]2 n/d 4 F(n/2,1,1) n q=1 d S q 1 s ( 1) q ([d 1]F(n/2d,1,1) + [d 3]F(n/2d,1,0)). n q=1 d S q B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and27 Constan / 34

28 Table: Different values of N(16, t 1, t 2, t 3, t 4 ), where t i = 0, 1. Case No. (t 1,t 2, t 3,t 4) our estimate N(16, t 1,t 2,t 3,t 4) 1 (1, 1, 1, 0) (1, 0, 0, 1) (1, 1, 1, 1) (1, 0, 0, 0) (0, 0, 1, 0) (0, 0, 1, 1) (1, 1, 0, 0) (1, 0, 1, 1) (1, 1, 0, 1) (1, 0, 1, 0) (0, 1, 1, 1) (0, 1, 1, 0) (0, 0, 0, 0) (0, 0, 0, 1) (0, 1, 0, 0) (0, 1, 0, 1) Total B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and28 Constan / 34

29 Table: Different values of N(18, t 1, t 2, t 3, t 4 ), where t i = 0, 1. Case No. (t 1,t 2, t 3,t 4) our estimate N(18, t 1,t 2,t 3,t 4) 1 (1, 1, 1, 0) (1, 0, 0, 1) (1, 1, 1, 1) (1, 0, 0, 0) (0, 0, 1, 0) (0, 0, 1, 1) (1, 1, 0, 0) (1, 0, 1, 1) (1, 1, 0, 1) (1, 0, 1, 0) (0, 1, 1, 1) (0, 1, 1, 0) (0, 0, 0, 0) (0, 0, 0, 1) (0, 1, 0, 0) (0, 1, 0, 1) Total B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and29 Constan / 34

30 Future Directions (1) A natural way to extend this problem is completing the counting argument to find N(n,t 1,t 2,t 3,t 4 ). (2) Studying the number N(n,t 1,t 2,t 3,t 4 ), where n is an odd number. (3) Change the finite field to a general field F q (4) Studying the number N(n,t 1,...,t k ), where 4 k n. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and30 Constan / 34

31 Thank You B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and31 Constan / 34

32 References L. Carlitz, A theorem of Dickson on irreducible polynomials, Proc. Amer. Math. Soc., 3 (1952), K. Cattell, C. R. Miers, F. Ruskey, M. Serra and J. Sawada, The number of irreducible polynomials over GF(2) with given trace and subtrace, J. Combin. Math. Combin. Comput., 47 (2003), S. D. Cohen, Explicit theorems on generator polynomials, Finite Fields and Their Applications, 11 (2005), S. D. Cohen, M. Presern, Primitive polynomials with prescribed second coefficient, Glasgow Mathematical Journal Trust, 48 (2006), S. D. Cohen, Primitive elements and polynomials with arbitrary trace, Journal of American Mathematics Society, 83 (1990), 1???7. B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and32 Constan / 34

33 R. W. Fitzgerald and J. L. Yucas, Irreducible polynomials over GF(2) with three prescribed coefficients, Finite Fields and Their Applications, 9 (2003), E. N. Kuz???min, On a class of irreducible polynomials over a finite field, Dokl. Akad. Nauk SSSr 313 (3) (1990), (Russian: English translation in Soviet Math. Dokl. 42(1) (1991), ) R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge, second edition, M. Moisio, Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm. Acta Artith., to appear M. Moisio and K. Ranto, Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed, Finite Fields and Their Applications, 14 (2008), B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and33 Constan / 34

34 J.L. Yucas, Irreducible polynomials over finite fields with prescribed trace/prescribed constant term. Finite Fields and Their Applications, 12 (2006), J. L. Yucas and G. L. Mullen, Irreducible polynomials over GF(2) with prescribed coefficients, Discrete Mathematics, 274 (2004), D. Wan, Generators and irreducible polynomials over finite fields. Math. Comp., 219 (1997), B. Omidi Koma () The Number of Irreducible Polynomials of Degree n over F q with JulyGiven 22, 2010 Trace and34 Constan / 34